Inspired by this question, which I realized I couldn't answer (because model theory and me don't get along).
I've made a few edits to (hopefully constructively) tighten the question a bit.
If for theories $T,T'$ it happens that $T\vdash Con(T')$, what does this really tell me about models of $T$ with respect to $T'$? Does it tell us that every model of $T$ has a definable substructure that's a model of $T'$, or is it more subtle? Or is it even interesting from a model theoretic standpoint (i.e. is the proof theoretic relationship between $T$ and $T'$ generally the only interesting one)?
I would like to assume $T$ and $T'$ can both yield PA by default, but I would be interested if anything general can be said about weaker theories.