A standard model of ZF need not be transitive, of course, and Wikipedia makes the believable claim that existence of a standard model does not imply existence of a transitive one.

As to proving that claim, 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?

Whether that is right or wrong, 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.