We study multilevel fermions in an optical lattice described by the Hubbard model with on-site -symmetric interactions. We show that in an appropriate parameter regime this system can be mapped onto a spin model with all-to-all -symmetric couplings. Raman pulses that address internal spin states modify the atomic dispersion relation and induce spin-orbit coupling, which can act as a synthetic inhomogeneous magnetic field that competes with the exchange interactions. We investigate the mean-field dynamical phase diagram of the resulting model as a function of and different initial configurations that are accessible with Raman pulses. Consistent with previous studies for , we find that for some initial states the spin model exhibits two distinct dynamical phases that obey simple scaling relations with . Moreover, for we find that dynamical behavior can be highly sensitive to initial intraspin coherences. Our predictions are readily testable in current experiments with ultracold alkaline-earth-metal(-like) atoms.