This doesn't make sense. A theory is inconsistent if and only if it has no models. I don't know what you mean by an "inconsistent model" here.

It straightforwardly means that I don't know very much model theory and interpreted my limited reading in an incorrect way. It appears that I mean a nonstandard model of Peano Arithmetic which is syntactically consistent but which fails to be omega-consistent. Only omega-consistent logics maintain an exact isomorphism between their Goedel Numberings of theorems and actual theorems, AFAIK.