Fix some countable language $\Sigma$, and some reasonable way of interpreting reals as $\Sigma$-structures with domain $\omega$. Let $T$ be a complete $\Sigma$-theory with continuum-many isomorphism types of countable models. Then of course there is a perfect set $P$ of reals, whose elements code pairwise nonisomorphic models of $T$.

However, it occurred to me recently that I see no reason why we should be able to get all the models of $T$ so represented. Specifically:

Need there be a perfect set $P$ of reals, such that every countable model of $T$ has exactly one real in $P$ which codes a copy of it?

I assume the answer is no - I imagine that models of sufficiently low Scott rank might be hard to incorporate into any such $P$ - but I don't see how to prove this.

If the answer is indeed no, what are the $T$s for which the answer is yes?

Fix some countable language $\Sigma$, and some reasonable way of interpreting reals as $\Sigma$-structures with domain $\omega$. Let $T$ be a complete $\Sigma$-theory with continuum-many isomorphism types of countable models (EDIT) which is not a counterexample to Vaught's conjecture. Then of course there is a perfect set $P$ of reals, whose elements code pairwise nonisomorphic models of $T$.

However, it occurred to me recently that I see no reason why we should be able to get all the models of $T$ so represented. Specifically:

Need there be a perfect set $P$ of reals, such that every countable model of $T$ has exactly one real in $P$ which codes a copy of it?

I assume the answer is no - I imagine that models of sufficiently low Scott rank might be hard to incorporate into any such $P$ - but I don't see how to prove this.

If the answer is indeed no, what are the $T$s for which the answer is yes?