Critical points in coupled Potts models and correlated percolation

We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a $q$-state Potts model coupled to an $r$-state Potts model in two dimensions. For integer values of $q$ and $r$ the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case ($q=r=2$). Since our results extend to continuous values of the number of states, we can access the limit $r\to 1$ corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin-Kasteleyn clusters by analytical continuation.