This is probably a pretty simple question, but I'm tying myself in knots over it.
We're all familiar with the Reflection Theorem, Lowenheim-Skolem Theorem, and Mostowski Collapse Lemma for getting countable transitive models of finite fragments of ZFC. You take a finite fragment $\Gamma$, use the reflection theorem to get that $\Gamma$ holds in some $V_\kappa$, Skolemise over $V_\kappa$, and then Collapse to a ctm.
My question. Is there (or could there be) a countable transitive model satisfying the same first-order truths as $V$? Obviously, by Tarski, such a countable transitive model is not first-order definable.
A possible route:
Suppose full second-order reflection is true of $V$. Let $A$ be a second-order parameter that contains a witness for every parameter-free sentence of first-order $ZFC$ true in $V$. Then, just reflect the sentence $x \in A$, Skolemise, and Collapse as normal (I feel like I might be pulling a fast-one here).
Well. There could be, or there might not be.
If there are no transitive models, then of course the answer is negative. But it could be positive, and in fact without evening increasing the consistency strength of $\sf ZFC$.
We augment the language of $\sf ZFC$ by adding a constant symbol $M$, and we add the following axioms:
If $\sf ZFC$ is consistent, then any finite fragment of this theory is consistent due to the reflection theorem. Simply find a large enough $\alpha$ such that $V_\alpha$ reflects whatever you wanted, and find an elementary equivalent countable transitive model.
If $V$ satisfies this theory, then $M$ is a countable transitive model and $\varphi\leftrightarrow\varphi^M$ holds there. So $M$ is in fact a countable model which is elementary equivalent to $V$.
(The entire thing is due to Feferman.)