# Existence of a transitive model is strictly stronger than consistency?

It seems like we should be able to prove that the existence of a transitive model for ZFC is strictly stronger than Con(ZFC), but I can't find anything saying so / giving an argument for it. Is there a standard way of demonstrating this?

An example of what I'm looking for, if it existed: given Con(ZFC), is there a way of generating a model which models Con(ZFC) but which believes that no models of ZFC are transitive?

Any model of ZFC+Con(ZFC)+$\neg$Con(ZFC+Con(ZFC)) should do. (This theory is, by the second incompleteness theorem, consistent if ZFC+Con(ZFC) is).