Examples of Phonologies that Break Charting Schemes

[Visions1]

I remember hearing a while back that there are some languages that don't play well with how linguists conceptualize and chart human sounds.
Famously, this would include Marshallese vowels from that one paper. You know, with the ☕︎?
Are there any other languages where the phones (consonant/vowel) don't range the ways our transcription systems are designed?

(Similarly, are there any examples where we lump the categories differently in a language than as standard - e.g. peripheral/coronal v.s. standard labial/coronal/velar?)

[Creyeditor]

What about american structuralist phonological analyses with all their weird archiphonemes? I think I saw a consonant inventory with only capital letters once

[Visions1]

Jungian psychology for linguists. Lol.
Could you link me some examples? I'd like to see how deep they can get.

[Nortaneous]

What's the phoneme inventory of Laze?

[Visions1]

That's not really what I'm talking about. Sure, there's allophony, but it fits our phoneme categorization schemes.
Marshallese, by contrast, breaks them. Look into it, you'll see.

[Salmoneus]

How on earth does Marshallese "break our phoneme categorization schemes"?

If you mean the vowels, it has four vowels, /æ ɛ e i/. Nothing scheme-breaking there.

It does have extensive allophony, so these are frequently not realised as [æ ɛ e i], but may be backed and/or rounded. Reflecting that, some people prefer to use the symbols for central vowels as the phonemic symbols. One guy, as you point out, even used random symbols, to encourage readers not to automatically assume that the front unrounded realisations were in some way primary. But of course the choice of symbols to describe phonemes is ultimately arbitrary, and says nothing profound about the way the phonemes themselves are categorized.

Likewise, you could multiply the vowels by not recognising the underlying glides that the normal analysis includes. Or, more radically, you could ignore the secondary articulations on the consonants that the normal analysis includes, and instead posit more vowels, shifting the allophony to the consonants. This is presumably considered less elegant for some reason (I don't know the details), but it's always possible. [we talk about the phonemes a language has, but really what it has are phonemic distinctions, and phonemes themselves are a notational convenience]

But both these issues - extensive allophony, and the possibility of multiple valid phonemic notations for the underlying distinctions - are present in all languages, so there's nothing scheme-breaking here. Marshallese just makes these issues more obvious than in most languages (as languages with small inventories often do).

[Visions1]

Well then, I'd like to see ones more on the extreme end.