Steinitz's theorem on the existence of an algebraically closed extension of every field clearly generalizes to broader model-theoretic contexts (existentially closed structures).
But there is another half to Steinitz's theorem that I had not thought about, which is the uniqueness of the algebraic closure of a field. I believe that it comes from the uniqueness of the splitting field extension of a field and a polynomial. Does this generalize to other contexts? That is: given an AE theory $T$ and $M \models T$, parameters $\overline a$ in $M$, a p.p. formula $\phi(\overline x, y)$ such that $T \cup \operatorname{Diag}(A) \cup \{\phi(\overline a, y)\}$ is consistent, does there exist a unique and minimal extension of $A'$ realizing $\phi(\overline a, y)$?
I'm guessing not, but unable to come up with an actual example. A bonus would be where $T$ is the AE theory of some structure, in which case there seems little room to maneuver.