= Solution
<Existential sentences> are preserved under extensions: witnesses remain present and quantifier-free truth is unchanged by an <structure embedding>[embedding]. Therefore any existential axiomatization implies the specified extension-closure property.
For <existential axiomatization from extension closure>, let $E$ be all existential consequences of $T$, and suppose $A\models E$. We claim that $T\cup\operatorname{Th}_\forall(A)$ is satisfiable. Otherwise compactness supplies finitely many <universal sentences> $\theta_1,\ldots,\theta_m$ true in $A$ such that $T\models\neg(\theta_1\land\cdots\land\theta_m)$. The finite <conjunction> is universal after renaming <bound variables>, so its <negation> is existential. It belongs to $E$ but is false in $A$, a <contradiction>.
The preceding universal-theory <structure embedding>[embedding] result now yields $B\models T$ and $A^*\equiv A$ with $B\subseteq A^*$. By the assumed extension closure, $A^*\models T$; <elementary equivalence> implies $A\models T$. Thus $E$ axiomatizes exactly the <first-order model>[models] of $T$, proving \b[existential axiomatizability is equivalent to closure under extensions]. This includes the <consistent first-order theory>[inconsistent] case, whose empty <first-order model>[model] class is axiomatized by an existential <contradiction>.
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