Shouldn't relativisation contain consistency proof ?
Whether or not $P^{A,B}$ equals $NP^{A,B}$ is going on the particular oracle languages A and B that you are using.  Iirc, in the BGS paper the language A is TQBF (or any other PSPACE-complete language).  The language B, somewhat ironically, is actually defined via a diagonalization construction.  Now, if you use the construction from the paper and apply it to TMs with an A oracle, then the resulting language B will be such that $P^{A,B} \ne NP^{A,B}$.  On the other hand, if you stick with the original language B from the paper, it will depend on the details of B's construction.  If B is in PSPACE (I don't know offhand if it is or not), then a TM with an A oracle could simulate any queries to B, and you should have $P^{A,B} = NP^{A,B}$.

The bottom line, though, is that this has nothing to do with consistency--it is just a technical question about the specific oracle languages A and B (and you could get different answers depending on the particular choices of A and B that you use).