Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?

You may assume that ZFC has transitive models. Note that $M$ is minimal iff $∀M' \, M'⊈M$ and minimum iff $∀M' \, M'⊇M$.

It may be tempting to consider ZFC + $0^\#$ (assuming large cardinal axioms), but while this theory has a minimum inner model (i.e. $L[0^\#]$), it has incomparable minimal transitive models. Model comparability uses iterability, but transitiveness does not suffice for iterability. Moreover, for every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^M = α < ω_1$, the intersection of all such $M$ equals $L_α$, and furthermore a subset of $L_α$ is definable (with parameters) in all such $M$ iff it is in $L_{α^{+,\mathrm{CK}}}$. To see this (briefly), $0^\#$ allows $M$ to 'continue' $L$ beyond $α$, and $L_{α^{+,\mathrm{CK}}}⊆(L_{α^{+,\mathrm{CK}}})^M$ (the well-founded part of any model of KP being admissible), and so $L_{α^{+,\mathrm{CK}}}∩V_α=L_α$. Also, existence of $M$ is $Σ^1_1(α)$, so the intersection of all $M$ is at most $L_{α^{+,\mathrm{CK}}}$.

A minimum transitive model of ZFC + $A$ for a *statement* $A$ cannot be produced through set forcing either. Also, I think that minimal models need not satisfy $V=HOD$; not sure about minimum models.

Thus, minimum transitive models usually either satisfy $V=L$ or do not exist. However, I suspect that exceptions exist, including in the form $L_α[r]$ with $r∈ℝ$, but require an interesting coding argument. Perhaps there is such an $r$ computable from the theory of the least transitive model of ZFC $L_α$, with $r$ self-verifying (in $L_α[r]$) and yet weak enough for $L_α[r]⊨\text{ZFC}$.

*Update:* I accepted Farmer Schlutzenberg's positive answer (see also the partial answers by Hamkins and Enayat). A remaining open problem is whether such a $T$ can be obtained by extending ZFC with a single statement.

minimumtransitive model. So an affirmative answer to my question would a fortiori imply an affirmative answer to yours, since levels of $L$ are comparable. (Maybe the following phrasing is better: say that a transitive set $A$ isbasiciff there is some c.e. theory $T$ such that $A$ is the minimum transitive model of $T$. Are the basic sets linearly orderd by $\in$?) $\endgroup$