Through the years, I have suggested some typologically unlikely, and maybe even some typologically impossible ideas in this blog. I find typologically unlikely - and even antiuniversal - systems somewhat interesting. However, I do believe there are some types of systems that we even find in some conlangs, which violate a type of constraint that I believe is a solid wall of impossibility.
In my own thoughts on this topic, I basically think of them just by the term "genuinely impossible systems". However, an issue with them is that their surface realization is possible - and there's probably multiple genuinely impossible systems corresponding to every possible surface realization.
Here's a phonological example. In antirealistic, there are two phonemes /b/ and /p/. These have the following realizations. NB: the phones themselves aren't really the interesting thing here, their relative realizations are:
initial: /b/ : [b], /p/ : [p]
medial: /b/ : [p], /p/: [b]
Why do I hold this to be unrealistic? Unless there's super-strong morphophonemic reasons to identify the [b] inside a word with /p/, and the [b] in the onset with /b/, I am very certain that any child or foreigner learning this language will identify the [b]-sounds as /b/, and the [p]-sounds as /p/. In lieu of a very strong morphophonemic relation here, there's no way a learner would identify them like that - even if the writing system maintained the identity.
A morphological example, then - and I don't think we find much of these in conlangs (unlike the phonological example seen above). In unrealistic, there are special verb forms corresponding to English -ing, and in unrealistic it's -int. However, for intransitive verbs, this consists of -i- (intransitive) and -nt (intransitive active participle), whereas for transitive verbs it consits of -in- (transitive) and -t (transitive active participle). Unless -in-, -i-, -n-, -t and -nt exist as independent morphemes but only ever occur in this context, there's no reason a learner would identify this as a complex suffix.
Syntax, then. Can anyone come up with a good syntactical example of a similar infeasible structure?