To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal consistency and the inner model hypothesis. Throughout, "ctm" means "countable transitive model of $\mathsf{ZFC}$," an inner model of a ctm $\mathcal{A}$ is a ctm $\mathcal{B}\subseteq\mathcal{A}$ with the same ordinals, and ctms are compatible iff they are each inner models of some single ctm.
Question
Given a class $\mathbb{K}$ of ctms, say that a ctm $\mathcal{M}$ satisfies the $\mathbb{K}$-IMH iff for every first-order sentence $\varphi$ and every ctm $\mathcal{A}\in\mathbb{K}$ which satisfies $\varphi$ and is compatible with $\mathcal{M}$ there is an inner model $\mathcal{B}\subseteq\mathcal{M}$ with $\mathcal{B}\in\mathbb{K}$ and $\mathcal{B}\models\varphi$. Intuitively, anything which holds in an inner model of an outer model of $\mathcal{M}$ already holds in some inner model of $\mathcal{M}$ after we appropriately restrict attention to models in $\mathbb{K}$. The classical IMH corresponds to $\mathbb{K}$ = $\{$all ctms$\}$ - call this "$\mathbb{A}$" for "all."
- Note that I'm not requiring $\mathcal{M}\in\mathbb{K}$ here; this is probably inessential, but I do think it is natural. Meanwhile I'm much more ambivalent about whether compatibility should also have to "go through $\mathbb{K}$," that is, whether I should replace "$\mathcal{A}$ is compatible with $\mathcal{M}$" with "$\mathcal{A}$ and $\mathcal{M}$ are common inner models of some ctm in $\mathbb{K}$;" I've tentatively decided not to include that restriction, but I'd be interested in answers which do adopt it as well.
I'm curious about what happens if we shift from $\mathbb{A}$ to the set of ctms $\mathbb{T}$ which don't "break" when equipped with their own elementary diagram. To be precise, suppose $\mathcal{M}$ is a ctm. Via any of the usual coding mechanisms, let $T\subseteq\mathcal{M}$ code the full elementary diagram of $\mathcal{M}$. Then $\mathcal{M}\in\mathbb{T}$ iff the expansion $(\mathcal{M},T)$ satisfies the Separation and Replacement schemes in the language expanded to include a predicate for $T$. If I have my jargon right, $\mathbb{T}$ is the set of ctms whose unique total satisfaction class is amenable (but the literature I could find on satisfaction classes doesn't seem relevant to this particular question).
My question, then, is:
What can we say about models satisfying the $\mathbb{T}$-IMH?
See below for a couple specific sub-questions.
Motivation
Say that a ctm $\mathcal{A}$ has a weird real iff there is a real $r\in\mathcal{A}$ such that no $\alpha\in\mathcal{A}\cap\mathsf{Ord}$ has $L_\alpha[r]\models\mathsf{ZFC}$. Beller-David showed that every ctm has an outer model with a weird real, and Friedman used this in his above-linked paper to show that $(i)$ no ctm satisfying the $\mathbb{A}$-IMH can have inaccessible cardinals and $(ii)$ the version of $\mathbb{A}$-IMH with real parameters is inconsistent. But it's also the case that Beller-David doesn't apply to $\mathbb{T}$-IMH: if $\alpha=\min\{\beta: L_\beta[r]\models\mathsf{ZFC}\}$, then $L_\alpha[r]\not\in\mathbb{T}$, and indeed no ctm which is "pointwise definable over a single parameter" is in $\mathbb{T}$. Moreover, $\mathbb{T}$ is (to me at least) a reasonably interesting class of models to consider from a foundational perspective, so this isn't entirely silly.
On the other hand, entirely defanging the IMH would sort of ruin the point. So I'm curious what details survive the shift from $\mathbb{A}$-IMH to $\mathbb{T}$-IMH. In particular:
Are there ctms satisfying the $\mathbb{T}$-IMH with inaccessibles?
Can there be a ctm $\mathcal{M}$ satisfying the $\mathbb{T}$-IMH for sentences with real parameters from $\mathcal{M}$?