Allowing another instructor to come in and signoff a student for a checkride retry, and having it count towards their pass rate, to me sounds like the FAA is opening themselves up to a lot of unfairness. Whether it's officially "allowed" or not, I have no idea.
When I first took my instrument checkride, I messed up the DME arc by setting up the GPS wrong. The DPE said basically "you just failed, do you want to continue, or just go back?" I said I want to continue. The rest of the flight went well, the partial panel approach was good, the holds were done right, etc. Basically all I had to do was jump in the sim, log 0.1 of "remedial training", and have my instructor do another 8710. When I redid my checkride, all we did was takeoff, join a 10 mile arc, fly it for about 2 minutes, then he said "you pass, lets head back"
The thought of another instructor benefiting from giving me that 0.1 of sim and re-doing my 8710, while my other instructor gets punished for all he did, just seems unfair. Anyways, whats to stop my instructor from "selling" my 8710 to another instructor so someone else can get a bump in their percentage? I've never been a fan of the idea of passrate being used as a measure of an instructor's ability, and now I'm an even less of a fan.