'Isomorphic' seems like an unfortunate choice, given that you want to allow for fundamentally different theories (I suppose). But then again, I'm not entirely sure that I get the point of your proposal.

@მამუკა ჯიბლაძე Yes, this is possible. Let $\phi \equiv M_1^{\sharp} \text{ exists } + \text{ there is no Woodin cardinal}$. (If $\delta$ is the least Woodin cardinal in our universe, then $V_{\delta}$ satisfies this theory, so it is not outright inconsistent.) Now there is no Woodin cardinal in our universe, but we can linearly iterate the top extender of $M_1^\sharp$ out of the universe, thereby obtaining the inner model $M_1$ which has a Woodin cardinal (from its point of view - the Woodin cardinal of $M_1$ is countable in $V$).

@მამუკაჯიბლაძე What I mean is that starting from $\operatorname{ZFC} + \phi$ for some sufficiently strong condition $\phi$ (say $\phi \equiv \text{Axiom of Determinacy}$) we often can't prove that certain large cardinals must exists (in fact, we can prove that we can't prove that), but - at the same time - we can produce a certain inner model $\mathcal M$ such that inside $\mathcal M$ there provably are large cardinals.

[...] A famous result along those lines (due to Woodin) is the equiconsitency of $\operatorname{ZF} + \operatorname{AD}$ and $\operatorname{ZFC} + \text{ there are infinitely many Woodin cardinals}$. It's important to note that from a certain assumption $\dagger$ we - in general - don't get large cardinals in our universe, but that $\dagger$ allows us to prove the existence of certain large cardinals in a (canonical) inner model. Typically these large cardinals are in fact countable in our background universe - and hence far from being 'large' in that sense.

@მამუკაჯიბლაძე In Gödel's constructible universe $L$, only so called 'small large cardinals' can exist. Therefore we always have a canonical inner model with relatively few large cardinals and moreover, any inner model $M$ of $V$ has an inner model with the same large cardinals, since $L^{M}$ (i.e. $L$ constructed inside $M$) is just $L$. On the other hand, not only do large cardinals imply models with other large cardinals - all kinds of axiom imply large cardinals in inner models. [...]

With Gabe's permission, I uploaded my notes here. He is also finalizing a paper on the subject, so keep an eye out for that. (Sorry about the bad handwriting - it was freezing cold inside the lecture hall.)

@Victoria My bad. This link should work.

I took notes during Gabe's talk yesterday and he indeed outlined of proof of $\operatorname{NBG} + \operatorname{DC} + \text{Reinhardt} \vdash \operatorname{Con}(\operatorname{ZFC} + \operatorname{I_0})$. The use of $\operatorname{DC}$ - according to Gabe - is unfortunate, but it's unknown how to avoid it for now. If Gabe is fine with it (I'll ask him later today), I'd be happy to share my notes in case anyone is interested in them.

This question over at MSE brought me here and reading your answer, I'm pretty sure that I know what you mean by "overspill": Since $M$ contains nonstandard naturals, there is some nonstandard natural $n$ (from $M$'s perspective) such that $M \models k < n$ for every standard natural $k$ and hence for all standard axiom of $ZFC$. You now apply the Reflection Principle in $M$ to get some $(V_{\alpha})^M$ such that all standard axioms are absolute between it and $M$. Is that right?