Take the theory $\sf ZFC + classes$ exposited in this posting. Now replace the axiom schema of replacement in non-class style by an axiom of limitation of size, that is: $$\forall \mathcal X \subset \mathcal V \ (\mathcal X \in \mathcal V \iff |\mathcal X| < \mathcal V)$$ where $\mathcal V$ is the class of all sets (i.e. objects of the first sort).
Now add the following homogeneity axiom schema: $$ \phi^1 \iff \phi^2$$, where $\phi^1, \phi^2$ are homogeneous in non-class, class style respectively, and each is the other formula re-written in its style.
Now clearly this theory call it $\sf ZFC + classes_3$ would interpret both $\sf ZFC + classes$ and $\sf ZFC + classes_2$. But here it is stronger since it clearly interprets $\sf MK$. This theory proves the existence of a class that is of inaccessible cardinality, and thus a set likewise, but this would further pump $\mathcal V$ to the next inaccessible and so there must be a set likewise, and so on...
Now is this homogeneity axiom acting in a way similar to reflection acting on top of limitation of size axiom, like the theory here? or is it weaker? or is it stronger acting like the resemblance axiom schema presented in this theory? or is it even inconsistent?