This was previously asked and bountied on MSE:
For brevity, let $T$ be $\mathsf{ZFC+V=L}$.
Say that an extension of $\mathsf{ZFC}$ is $\omega$-complete iff it has exactly one $\omega$-model up to elementary equivalence. While the $\omega$-incompleteness of $T$ is easily provable in theories only slightly stronger than $T$ itself, I don't immediately see how to do it in $T$ alone. My question is:
Is the theory $S:=T+$ "$T$ is $\omega$-complete" consistent?
Here are a couple observations:
If we replace "$\omega$-model" by "well-founded model," the answer is obviously yes under standard assumptions. Let $\alpha$ be the second-smallest ordinal such that $L_\alpha\models\mathsf{ZFC}$. Then $L_\alpha$ also satisfies "$\mathsf{ZFC+V=L}$ has exactly one well-founded model." Unfortunately, we have no analogous hierarchy of $\omega$-models, so this is a non-starter here.
As to the specific choice of theory in question, the point is that (something like) $\mathsf{V=L}$ is needed to block an easy proof of a negative answer via forcing. For example, reasoning in $\mathsf{ZFC}$, if $\mathsf{ZFC}$ had an $\omega$-model $\mathcal{M}$ it would have a countable one $\hat{\mathcal{M}}$, and we could force over $\hat{\mathcal{M}}$ to get a non-elementarily-equivalent $\omega$-model $\hat{\mathcal{N}}$. (Forcing over ill-founded countable models is no harder really than forcing over well-founded ones.) The key point here is that forcing preserves $\mathsf{ZFC}$. This breaks down of course for $\mathsf{V=L}$ and so this argument is irrelevant here. Given the paucity of techniques we currently have for building models of $\mathsf{ZFC}$ in the first place, this seems to be a real issue.
Ultimately I suspect that the answer is negative, but the above two points between them rule out all the lines of attack I've been able to think of so far.
EDIT: In light of Farmer S's answer below, let me explicitly mention a rule of thumb which I forgot: when thinking about properties which are not too far from first-order definable, always consider the hyperarithmetic hierarchy!
For example, for every $\mathcal{L}_{\omega_1,\omega}$-sentence $\varphi$, if $\varphi$ has a model then it has a model $M$ which is countable in $L$, and moreover the $L$-least (real coding a) model of $\varphi$ is hyperarithmetic relative to (any real coding) $\varphi$. The property "Is an $\omega$-model of $T$" is expressible as a computable $\mathcal{L}_{\omega_1,\omega}$-sentence, and this drives Farmer S's point that $T^+\in L_{\omega_1^{CK}}$.