We present a generalized multiferroic phase diagram for orthorhombic RMnO₃(R=Gd-Lu) based on coherently grown thin films. The magnetic order was identified by neutron-diffraction and resonant soft x-ray scattering experiments. For large R-ions (R=Gd-Dy), the transition temperature to a long-range ordered antiferromagnetic phase is only weakly dependent on the R-ion radius, but decreases monotonically with decreasing R-ion radius for films with R=Ho-Lu. The antiferromagnetic phase is characterized by an incommensurate order of the MN³⁺ spins, which successively locks into a commensurate E-type state. These findings confirm a uniform multiferroic ground state independent of the R ion and are in excellent agreement with predicted properties of strain-induced multiferroicity in these materials. In particular, strong variation of multiferroic properties in these epitaxial films compared to bulk highlights the tuning ability of strain.