  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.
