Transitive models of $V≠L$ within L Suppose $V=L$. Can there be transitive models of $ZFC+V≠L$?
Let $M$ be a transitive model of ZFC. If $x\in M$, then $x\in L_\alpha$ for some $\alpha$ because $V=L$, but it's not evident to me that $\alpha\in M$.
Such an $M$ would necessarily have to be a set, since the only inner model is $L$ itself.
 A: Yes, transitive models inside $L$ can be very non-$L$-ish.
Specifically, recall Shoenfield absoluteness. Since "$T$ has a countable transitive model" is $\Sigma^1_2$,$^*$ by applying Downwards Lowenheim-Skolem in $V$ we have that whenever $T$ is a theory with a transitive set model in $V$ then $T$ has a countable transitive model in $L$.
So, for example, if $\mathsf{ZFC+\neg CH}$ + "There is a proper class of supercompacts" has a transitive model, then it has a constructible transitive model, despite the fact that of course that theory is highly incompatible with the axiom of constructibility for both large cardinal and combinatorial reasons.

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*It may help, visualization-wise, to consider e.g. $L_{\omega^2}(\mathbb{R}^L)$. This is a transitive set contained in $L$ of height $\omega^2$ but containing all constructible reals - basically, it's "short and wide" in a way which makes it very different from any level of $L$. Now that's sort of a bad example since it's "informationally" equivalent to the better-behaved $L_{\omega_1^L}$ - each $L_\alpha$ for $\alpha<\omega_1^L$ is represented by a real in $L_{\omega^2}(\mathbb{R}^L)$ and can be "decoded" in a definable way - but it's a good first taste of how the particular shape of the levels of $L$, rather than their mere constructibility, constrains their behavior.


EDIT: There's another theorem which is relevant here. It gets a weaker conclusion than Shoenfield, but is quite different and interesting:

(Barwise) Every countable model of $\mathsf{ZF}$ has an end extension which is a model of $\mathsf{ZFC+V=L}$.

See here. Of course, that end extension will probably be ill-founded - that's why Barwise's theorem doesn't prove $\mathsf{V=L}$ outright. Incidentally, the picture at that blogpost is quite nice on its own - note the added width, in addition to height, per the comment about the shape of levels of $L$ in the previous section. Barwise's theorem does not give us a top extension.
Barwise's theorem lets us transfer consistency results: if $\mathsf{ZFC}$ + "There is a transitive model of $T$" is consistent then so is $\mathsf{ZFC+V=L}$ + "There is a transitive model of $T$." More generally, note that end extensions preserve internal transitivity satisfaction: if $M\models\mathsf{ZF}$, $A$ and $T$ are in $A$, $M$ thinks $T$ is a theory and $A$ is a transitive set satisfying $T$, and $N$ is an end extension of $M$ (perhaps one satisfying $\mathsf{ZFC+V=L}$!), then $N$ also thinks that $A$ is a transitive model of the theory $T$.

$^*$OK, that's not strictly true: rather, it's $\Sigma^1_2$ relative to $T$. So really all we can conclude is that every $\{\in\}$-theory which is in $L$ which has a transitive model in $V$ also has one in $L$.
A good example of how this can play out is to consider the following. Let $T_0=\mathsf{ZFC}$ + "$0^\sharp$ exists," and let $T_1$ be $T_0$ + axioms correctly stating each bit of $0^\sharp$. Now per the above (under reasonable hypotheses) $T_0$ has a transitive model in $L$. On the other hand, $T_1$ definitely won't: a transitive model of $T_1$ has to compute $0^\sharp$ correctly, which $L$ can't. But this is fine, since $T_1$ itself computes $0^\sharp$: $T_1\not\in L$ so we can't apply Shoenfield.
A: Let me add something to Noah's nice answer. If there are transitive set models of set theory, then there are such models of $V=L$, and therefore there is  a countable $\alpha$ such that $L_\alpha$ is a model (by the Löwenheim–Skolem theorem and condensation). Since $L_\alpha$ is countable, for any forcing poset $\mathbb P\in L_\alpha$ there are (in $L$) $\mathbb P$-generics $g$ over $L_\alpha$. Except for trivial cases, the resulting models $L_\alpha[g]$ all satisfy $V\ne L$ (although they all live in $L$).
This is actually a useful observation. Jensen exploits to great advantage a version of it in his proof of the theorem that for any countable sequence $\langle \alpha_\nu:\nu<\delta\rangle$ of countable admissible ordinals there is a real $x$ such that the $\alpha_\nu$ are precisely the first $\delta$ admissible ordinals over $x$.
It also shows up in other situations. For instance, in certain recursive definitions of Suslin trees in $L$, at certain key limit ordinals $\delta$ the construction chooses how to continue the partial tree $T_\delta$ built so far by picking a branch generic for $T_\delta$ over a model $L_\alpha$ whose $\omega_1$ is $\delta$.
