Equal-angle elastic scattering with unequal masses

Equal laboratory recoil angles constrain outgoing momenta, not equality of masses. In units $c=1$, take an incoming mass $m$ with energy $2m$ striking a stationary mass $2m$. The final momenta $m(13/8,\sqrt{3}/2,\sqrt{57}/8,0)$ and $m(19/8,\sqrt{3}/2,-\sqrt{57}/8,0)$ conserve four-momentum and have mass shells $m^2$ and $4m^2$. Both trajectories make angle $\arctan (\sqrt{19}/4)$ with the incident axis. This is a concrete counterexample to inferring equal masses from equal outgoing angles alone.