Existential axiomatization from extension closure

A first-order model class axiomatized by a theory is closed under extensions exactly when it has an existential axiomatization. If $A$ satisfies all existential consequences of $T$, then $T$ is jointly satisfiable with the universal theory of $A$: any finite obstruction would negate a universal conjunction and produce a violated existential consequence. Embed a model of $T$ into a model elementarily equivalent to $A$, apply extension closure, and transfer truth back to $A$.