This is all very well, but I think the important point to note here is that "texts" is an extremely annoying word, especially when used in exams in school when some woman from the exam board is saying it, repeatedly, on a tape played in a knackered old cassette player. I bet they try to get the phrase "mixed texts" into those tapes whenever they can, just to be extra annoying. Seethe!
Wow! "Sixths" definitely fits the bill. It was so obvious, and unlike "texts" it is a completely native word. I suppose what I meant in the original question was four consecutive obstruent phonemes--no nasals or liquids.
I think the "t" and "s" in "texts" are separate sounds in this instance, but I can see that they could be considered a single affricate.
If "glimpsed" and "jinxed" are used as nouns, then there are a couple of rather awkwardly contrived but still legitimate possessives that also have four consecutive obstruent phonemes: "The jinxed's thoughts were obliviously optimistic."
Thanks for the comments and thanks to sarra for the additional words.
Weebles wobble without falling down, so I'm not sure if by wobbliness you mean that there's something or nothing to my argument. All us Wordies kick linguistic butt on occasion, and get our butts kicked on others, so while I'm awaiting linguistic deathspeak...
Yes, there can be two breath breaks in "texts", but the first is optional whereas the second is hard to avoid. If tsetse were spelled tstse would it be one syllable or two? Also, can't "ts" be a single phoneme? What if I represent "xts" as "ksצ" (using a Hebrew letter tsadi) instead of ksts?
I don't think I could explain what constitutes a syllable (q.v. for an explanation in a day or two, perhaps) without breaking into linguist-speak of death — but to understand the wobbliness of your theory, note that there are actually two "breaks" of the breath (stops, which you could hold as long as you like) in this word, viz.: te—xt—s!
I'm not conversant with the conventions of judging such things, but seems to me "texts" is not quite one syllable. The breath is broken into two segments in the transition from "xt" to "s". I bid one and half syllables.
I mean four consecutive consonant phonemes (excluding liquids, like r, l, and w) if you spelled the word in IPA. Texts has ksts. "Strengths" has two consonants plus a liquid at the beginning and three consonant sounds at the end. The four letters at the beginning of "schmooze" actually represent only two phonemes. Aside from "Hoechst's" and maybe some similar possessives of words that are originally foreign, I don't think there's another word in English with four consonant sounds in a row in the same syllable.