A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to https://mathoverflow.net/questions/12804/large-cardinal-axioms-and-grothendieck-universes?rq=1 gives Tarski sets as an interesting example.

I should clarify since the terminology is not entirely standard.  By a standard model I mean what Wikipedia does, and what Joel David Hamkins does in his answer to https://mathoverflow.net/questions/124720/standard-model-of-zfc: the set membership relation $E$ of the model is the actual set membership relation $\in$, restricted to sets in $M$.  

Does existence of a standard model imply existence of a transitive one?  Existence of a Tarski set does imply existence of a Grothendieck universe.  But am I right to suspect that if there is a standard model, then the ordinals in the minimal model are not a true initial segment of the ordinals, so the minimal model is not transitive, and we can cut down to where that minimal model is the only standard one?  If I understand Guest289 correctly, his argument shows my guess is wrong, since a minimal model is transitive.   Do I understand that correctly?  Or does this come back to unclarity abut what is a standard model?

Anyway, does existence of a transitive model have higher consistency strength than existence of a standard model?  I would not be surprised if the axiom of choice lays a role here but I do not know if it does.