Probably stupid question: what happens in your question if we replace “countable transitive model” by just “transitive model”? Is there a statement $\tau$ such that a transitive model $M$ of ZF satisfies $\tau$ iff there is a transitive model $N$ of ZFC containing $M$ with the same ordinal height as $M$? Did you add “countable” because the answer without “countable” is trivial (I don't see it)?
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