I want to comment that both AD and "$V$ is $L$-like" are *consistent* with the existence of inner models of very strong theories. 

Assume there is a measurable cardinal $\kappa$ with a supercompact cardinal below it. Then $M_1^\#$ (the minimal active $(\omega,\omega_1+1)$-iterable $\omega$-sound mouse with a Woodin cardinal) exists and is fully iterable. Suppose $N$ is an inner model in which $\delta^{+M_1}$ is countable where $\delta$ is the Woodin cardinal of $M_1$. Then $N$ has an inner model with a supercompact cardinal. First, one can do a genericity iteration to make $V_{\kappa+3}$ set-generic over an iterate of $M_1$ for the extender algebra. As a consequence, in $M_1$, there is a condition in the extender algebra at $\delta$ that forces that there is a measurable cardinal $\bar \kappa < \delta$ such that $V_{\bar \kappa}$ has a supercompact cardinal. Since $\delta^{+M_1}$ is countable in $N$, $N$ can build a forcing extension $P$ of $M_1$ below this condition. Iterating away a measure on $\bar \kappa$ in $P$ yields an inner model of $P$ with a supercompact, and this inner model is contained in $N$. Similarly, $N$ will have an inner model of PFA (or MM). This shows, for example, that under large cardinal hypotheses, $L[M_1^\#]$ has an inner model of MM (and therefore so does $L(\mathbb R)$).

Therefore, we *can* in fact produce extensions of the theories ZF + AD and ZFC + "$V$ is $L$-like" that are consistent relative to large cardinals and that prove the existence of inner models of very strong theories, although these extensions are totally ad hoc: e.g., "ZF + AD + there is an inner model of ZFC with a cardinal that is supercompact to a measurable." This is closely analogous to the fact that assuming large cardinals, $L$ has a countable transitive model containing a supercompact cardinal.