I had to think for a while while writing that (oh wow hello awkward words) about that case specifically; genitive/possessive nouns.
The thing is, if you have let's call it gender in the first place, then both nouns are going to have their own specific genders. In order for one noun to cause the other to inflect according to the first's gender, you'd basically be needing to replace the gender of the second noun with the gender of the first noun.
e.g. for ease of example, if you have girl = feminine and shoe = neuter, and you have the girl's shoe, either the girl is going to become neuter or the shoe is going to become feminine.
Which you could do! It would be confusing and I'm pretty sure no existing languages do that, but it could be fun to experiment with.
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The thing is, if you have let's call it gender in the first place, then both nouns are going to have their own specific genders. In order for one noun to cause the other to inflect according to the first's gender, you'd basically be needing to replace the gender of the second noun with the gender of the first noun.
e.g. for ease of example, if you have girl = feminine and shoe = neuter, and you have the girl's shoe, either the girl is going to become neuter or the shoe is going to become feminine.
Which you could do! It would be confusing and I'm pretty sure no existing languages do that, but it could be fun to experiment with.