Much like verbs, we could have quirky-case adjectives. A quirky case adjective would, in a construction with a copula, require the noun to be in some atypical case - e.g.
me-dat is tired
We find that e.g. German sort of does this with kalt and such, but with kalt in particular, the meaning is different depending on the case of the ~subject, whereas the thing I am going for is simply that these adjectives always have quirky subject things going on.
What other things would we expect from such an adjective? I would posit that lack of congruence would be probable, and maybe that some comparative structures would be missing due to this lack of congruence.
Let's, however, imagine something even weirder to happen: when used as attributes, these adjectives start requiring pronouns that mark the case that the adjective triggers, but that agree in animacy (or gender or whatever) with the noun:
tired he*-dat man
the tired man
* here, "he" should only be seen as 'human', not as 'masculine' for now.
We go on one step further, and use indefinite pronouns with non-definite nouns, and regular personal pronouns with definite nouns:
tired one-dat man
a tired man
tired he-dat man
the tired man
As time goes by, these are merged into the adjective, but slightly worn so e.g. number morphemes are lost:
ADJ-[def/indef × animacy]-[animacy × case]
Suddenly, we have this situation:
Regular Adjective Quirky Adjective with Copulas: Number-Animacy Congruence yes no Adjective forces case marking on subject NP no yes as attributes in NPs: Number Congruence yes no** Animacy Congruence yes yes Case Congruence (optional*) no
Adjective has case marking, without correlation to case marking of NP no yes
* here, it's up to the conlanger really, whether they want case congruence or not. I like case congruence a lot so I'd go for it, but to each his own. ** as can be seen below, while writing this post I slightly changed my mind on number congruence for quirky case adjectives; however, I still find the idea of not having it rather appealing.
So, now we have a strong/weak adjective dichotomy much like in the Germanic languages, but only for a subset of the adjectives. The origin of the number and animacy congruence markers in the regular and quirky adjectives is historically different - granted, they may be cognates, but the regular adjective markers have probably been worn down a bit since being grammaticalized - so at most we'd expect similarity rather than formal identity between these markers. (This holds even if we don't wear down the number-morphemes of the dummy pronouns, similarity rather than identity still is to be expected.)
One further thing one could consider with these adjectives is restricting their use in causative constructions, and mayhap some other "not entirely trivial" circumstances?