On a whim I have started reading the Oxford Handbook of Historical Phonology. At about two and a half chapters in I have finally reached some discussion of practical questions in some detail, and the first claim to have struck me as empirically interesting is that “primary split can also reduce an inventory”.
For those not up to speed with or just not recalling the lingo (this is after all one of those terrible user-unfriendly terminological conventions along the lines of “type 1 / type 2 error“), I remind that a “primary split” is a conditional sound change that creates a sound (or rather, phoneme) already present in a system, contrasted with a “secondary split”, a conditional sound change that creates a sound not previously present. (I would advocate for using the more descriptive terms “split with merger” and “split without merger.)  They are distinct from a simple unconditional merger, or for that matter, from an unconditional non-merger.  Any particular change can fall under any one of these depending on the language.
My first thought were cases such as the fate of the labiovelar stops in Greek. Depending on their environment, these are reflected as any of the three “basic” stops (e.g. *kʷ > /p/, /t/, /k/; similarly for Proto-Greek *kʷʰ and *gʷ), and hence they ultimately disappear from the phoneme inventory. This kind of a situation does not seem to really show that primary split could eliminate a segment from a language’s inventory, though. Although any one of the changes could be stated conditionally, in reality one of the three changes must be the most recent chronologically — and at this point this change is then no longer a conditional sound change, but simply an unconditional merger. (I believe this status belongs to *kʷ > /p/. ) A similar sleight-of-hand could be really pulled whenever a sound eventually develops into multiple different reflexes: phonological inventories only offer a finite number of relevant environments, and even if there in fact is a default reflex, it can be also stated in terms of a set of particular environments. E.g. the development of PIE labiovelars in Indo-Iranian or Slavic could be stated roughly as “palatals before front vowels, velars before consonants and non-front vowels”, although only the former development is conditional, the latter instead unconditional; and indeed even feeding into, rather than independent of, palatalization before front vowels. (I.e. *Kʷ >> *Č / _E is, properly speaking, not a sound change but a sound correspondence, consisting of (at least) two sound changes: *Kʷ > *K followed by *K > *Č / _E.)
But it turns out the claim is actually something simpler. Proposed in a 2012 article “Primary split revisited” by Robert Blust, the idea is instead: if the segments involved are subject to positional constraints, afterwards it may be now possible to analyze either one of them as being now an allophone of something else. (He also passingly considers exactly the same example of labiovelars in Greek, with citation to a 2000 textbook by Sihler, but without noticing the flaw from chronology.) So the actual sounds involved do not disappear from a language’s phonology; they merely now end up in a complementary distribution, and the number of phonemes can be argued to have fallen. Certainly this should be possible.
Curiously, Blust presents this analysis as only a theoretical exercise, and ends up unable to propose any actual examples of the phenomenon. Google also tells me that Blust’s term “reductive primary split” still finds no additional hits out there. I take it upon myself to therefore offer a few examples.
1. Loss of *ŋ in Proto-Finnic
Proto-Uralic had an inventory of four nasals, *m *n *ń *ŋ. The Finnic branch has however reduced this inventory to just two, *m *n.  The fate of the palatalized nasal *ń has been simple, merging into *n (I believe with some vowel-coloring effects word-medially; but this is tangential to the point). The fate of the velar nasal *ŋ is more diverse. The most typical intervocalic reflexes are zero (with lengthening of the preceding vowel) and *v, presumably thru earlier *w; in consonant clusters, *Cŋ > *Cv, *ŋC > *uC, both presumably again thru *w. I would additionally posit an even earlier intermediate stage *ɰ behind both the zero and *w reflexes.
One exception to all this is found: the cluster *ŋk, surviving phonetically intact into Proto-Finnic and indeed into the modern Finnic languages. Phonologically looking, however, it would seem that there has been a change here as well. *[ŋ] cannot be reconstructed for Proto-Finnic in any other environment, and hence we now have reason to interpret [ŋk] as /nk/ (or if we really wanted, /mk/, or even /Nk/ with a neutralized placeless coda nasal). Thus the splits-with-merger *ŋ > ∅, *ŋ > *w and/or *ŋ > *ɰ have been reductive: even though they leave some instances of [ŋ] unscathed, */ŋ/ as a contrastive phoneme is still lost. All of this has been already noted at least as early as by Posti (1953).
This reductive primary split also in fact functions somewhat differently from Blust’s toy example. He suggests an example of a language contrasting /t/ and /s/ only before /i/, showing elsewhere only [t]; if, then, [ti] shifts to [si], the result will be the loss of this contrast — thus yielding /ti/ rather than /si/. In Finnic, it is however not the contrast *ŋ | *w that ends up lost; and what allows the final phonological reanalysis is not the earlier distribution of either of these consonants,  but rather the limited distribution of the “third wheel” consonant *n, which earlier did not occur in the position before *k.
2. Loss of *ɣ in Proto-Ugric
A reductive primary split that does function similarly to Blust’s example might be also found in Uralic. The current conventional reconstruction of Proto-Uralic includes a rare consonant *x, occurring only intervocally (and when followed by 2nd-syllable *ə, though this proves to be inessential to the point). Its reflexes across Uralic point towards a velar obstruent of some sort, though it does seem to have been distinct from *k, the other, well-established velar obstruent. We also find that the reflexes of *x and intervocalic *k indeed coincide to a large extent across Uralic. In some cases, the reflexes are inconveniently either zero (thus Permic, Mari, Samoyedic) or merged with something else still (thus Mordvinic). Here we cannot clearly rule out the option that it is *x that is first unconditionally lost or merged, followed by *k along the same trajectory only later. A merger to a distinct velar reflex *ɣ can be however found in the two Ob-Ugric language groups, Mansi and Khanty. The third Ugric language, Hungarian, has been proposed to also have passed thru a similar stage. If we suppose *[ɣ] was indeed the original sound value of “*x”, we would seem to have here exactly Blust’s situation: the contrast *ɣ | *k originally occurred only intervocally, and therefore the result of an intervocalic lenition of *k to [ɣ] will be counterintuitively phonemically */k/, not */ɣ/.
This situation would have been quite temporary, though: in all three Ugric groups, degemination of *kk probably introduced quite soon a new medial *k, leaving *ɣ again (?) a contrastive segment of limited distribution. At least the apparently parallel degemination of *pp, *tt however cannot be reconstructed for Proto-Ugric: it must be preceded by the lenition of medial *p and *t in Hungarian, which yield modern v, z, presumably thru intermediate *b, *d > *β, *ð. In Ob-Ugric by contrast *p, *t remain without lenition. Thus probably also *kk still remained in Proto-Ugric; and in any case degemination of *kk must postdate at least the lenition of single *k.
Of course a worse problem still is that the analysis depends on not particularly certain details of PU reconstruction. If *x was not *[ɣ] but something else, like *[x] or *[q], it would have been possible that it is intervocalic *k that first lenites to *[ɣ], after which *x simply unconditionally merges with this new allophone of *k.
But we can perhaps try again:
3. Loss of *ɣ⁽ʷ⁾ in Khanty varieties
The saga of *ɣ continues further in Khanty, with some rather similar development as in the previous case. From here on, the contrast with *k seems to be generally maintained (though we do find both of them giving /χ/ as a conditional reflex in Western Khanty). Instead it is the contrast *w | *ɣ that trends towards neutralization. One example could be found in Eastern Khanty, where intervocalic *w develops to *ɣʷ; and *ɣ splits, at least in the Surgut dialect group, to [ɣ] ~ [ɣʷ], the latter following most (but not all) Proto-Khanty labial vowels. We have some reason to consider the latter change older than the former: it is shared also with Western Khanty (with further *ɣʷ > /w/) and it could be reconstructed as an allophonic change already for Proto-Khanty. If so, *w > *ɣʷ in Proto-Eastern Khanty would be a reductive primary split: its result will be that *[ɣ] ~ *[ɣʷ] are now in a complementary distribution with word-initial *[w], and therefore they can be considered allophones of a single phoneme.
This situation, however, is not reflected as such in either of the two main branches of Eastern Khanty. In Surgut Khanty, mergers such as *ü > /i/ have now left /ɣ/ distinct from /w/ (= [w] ~ [ɣʷ]); in Far Eastern (Vakh-Vasjugan) Khanty, medial *p has been lenited to a new [w], while my proposed intermediate *[ɣʷ] has lost its labialization, likewise leaving /ɣ/ a clearly distinct phoneme from /w/. It would be also possible to suppose that *[ɣʷ] actually occurred in Proto-Eastern Khanty only as a medial allophone of /w/, and later [ɣʷ] as a reflex of *ɣ is an innovation of Surgut Khanty in particular, perhaps only at most areally connected with Western Khanty. Something like this is indeed suggested by Proto-Khanty roots of the shape *PÜɣ- (with a bilabial initial and a front rounded vowel) — these give /PIɣʷ-/ as expected in Surgut Khanty, but in several varieties of Western Khanty (Southern Khanty, transitional South/North dialects of Nizjam and Šerkaly), cheshirization to *ɣʷ > *w either fails to take place or is reverted, giving instead /PIɣ-/. Similarly, Proto-Khanty *-ăɣ- (probably with *ă being labial [ɒ̆]) gives Surgut /-ăɣʷ-/ but Western *-oχ-. Here too we don’t seem to have much evidence of a common Proto-Khanty development to *ɣʷ, and we should probably assume a separate labialization in Surgut (though something like *ɣʷ > *ʁʷ > *χʷ > /χ/ is at least theoretically conceivable).
The traditional reconstruction of Proto-Khanty (see e.g. Honti 1999: 75–77), actually goes even further and does not recognize distinct medial *w at all. In such a system, *ɣ would appear to have been an allophone of /w/ already at this point. This though implies positing a conditional merger *w > *[ɣ] already between Proto-Ugric and Proto-Khanty, which itself will be then a reductive primary split. — But I do find it preferable to assume that Proto-Uralic and Proto-Ugric *w was simply maintained as distinct all along in Western Khanty, especially since it seems to be possible to identify minimal pairs; one is Southern /sŏw/ < *sŏw ‘pole’ vs. /sŏχ/ < *sŏɣ ‘skin’.
I could probably think of several further examples of reductive primary splits in various languages — these have simply been the first three examples to come to my mind straight away. I can easily agree with Blust that perhaps this theoretical possibility has gone so far unrecognized due to an overreliance on just a few canonical examples mostly from Indo-European in discussions of the typology of sound change.
Honti, László. 1997. Az ugor hangtörténethez. Az ugor alapnyelv kérdéséhez: 31–39. Budapest.
Honti, László. 1999. Az obi-ugor konszonantizmus története. Szeged.
Posti, Lauri. 1953. From Pre-Finnic to Late Proto-Finnic. Finnisch-Ugrische Forschungen 31: 1–91.
 The “primary” / “secondary” terminology moreover seems to me to be kind of backwards. “Primary” splits appear to be unnecessary to assume as a separate phenomenon on the phonetic level at all, since it seems to me they can be always modelled as a series of two sound changes: a “phonetic secondary split”, followed by an unconditional merger of the newly created allophone.
 I have not seen this fourth option identified often, but it seems to be appropriate for any sufficiently advanced “phonetic drift”, taking a segment so far off-field that it cannot be identified anymore with its original phonological value. E.g. although no conditioning or merger has been necessarily involved, it would seem to be not at all appropriate to characterize West and Northwest European [ʁ] or [ʕ] as either a trill or a coronal, despite its origin from earlier /r/ (of course, meaning those varieties where a guttural fricative is the typical realization — when we do still find intermediate [ʀ], we could at least argue that this is the target realization and [ʁ] realizations are merely speech errors).
 The bilabials /p pʰ b/ are the result before a consonant, as well as before the “noncoloring” vowel /a/ and the “weakly labial” vowels /o ɔ/, i.e. environments where there is not much motivation for a conditional development. The dentals /t tʰ d/ are instead triggered by following front vowels /i e ɛ/ (assimilation), the velars /k kʰ g/ by a following or preceding close labial /u/ (dissimilation).
 The /ŋ/ encountered in modern Finnish is a later development, primarily by consonant gradation from *ŋk, later reinforced by loanwords. The native origin still leaving the interesting trace that singleton ˣ/-ŋ-/ remains foreign; only the geminate /-ŋː-/ is found intervocally. Similarly, /nʲ/ in modern Eastern Finnish, Karelian, Veps etc. arises by secondary palatalization, most widely thru apocope of *i.
 *ŋ does have a more-limited-than-average distribution in PU, being barred from the word-initial position. However, nothing in the analysis would change if we assumed that there did exist a word-initial *ŋ- that likewise changed to *w > *v in PF.