This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

Non-OP edit: To clarify, the question is whether the finite $\mathsf{FOL}$-axiomatizability of $Th_\mathsf{FOL}(C)$ necessarily implies the finite $\mathsf{FOL_{w/o=}}$-axiomatizability of $Th_{\mathsf{FOL_{w/o=}}}(C)$, regardless of whether the $\mathsf{FOL}$-deductive closure of $Th_{\mathsf{FOL_{w/o=}}}(C)$ coincides with $Th_\mathsf{FOL}(C)$ or not. (That is, we don't care whether $Mod(Th_{\mathsf{FOL_{w/o=}}}(C))\not=C.$)