I once had this on a Mazda RX8 I owned.
I had arranged to part-ex the car, so it was going in a few days, then typically, I got a blo*dy puncture in one of the rears.
If I remember rightly, the car had Bridgestone Potenza`s on already, but rightly or wrongly, (as you do), I wasn`t going to spend money on another Bridgestone only to sell it in two days time, so I duly put some sort of budget tyre on.
I couldn`t believe the difference !
Even in a straight line, you could really feel the car trying to squirm at the back, and as the Mazda is RWD, you could feel it even more-so.
I wouldn`t have believed an "alien" tyre could have made such a cock-up of the handling.
Fickle little buggers, are tyres............