Let $\cal U$ be a saturated model of inaccessible cardinality $\kappa$. For arbitrary $\cal D\subseteq U$ denote by $\langle\cal U,D\rangle$ the expansion of $\cal U$ with a new predicate for $\cal D$. Write $e({\cal D}/A)$ for collection of subsets $\cal C\subseteq U$ such that $\langle{\cal U,C}\rangle\equiv_A\langle{\cal U, D}\rangle$.
Question: Is there some easy condition on $e(\cal D)$ that characterizes sets that are externally definable?
Recall that $\cal D$ is externally definable if it has the form $\big\{a\ :\ \varphi(x;a)\in p\big\}$ for some global type $p\in S_x(\cal U)$ and some formula $\varphi(x;z)\in L$.
By the way of example, notice that we can easily characterize definablity. Namely, $\cal D$ is definable if and only if $e({\cal D}/A)$ is finite (or, for that matters, $=1$ or $<\kappa$) for some $A$.
Not relevant to the question (but motivates me): it can be shown that $\cal D$ is externally definable if for some $A$ the VC-dimension of $e({\cal D}/A)$ is finite. Clearly, the converse may fail unless we assume that $T$ is nip.
