We describe three-dimensional magnetohydrodynamical modelling of powerful radio galaxies in realistic poor cluster environments. This modelling extends our earlier work on the hydrodynamics of radio galaxies as a function of their cluster environment to consider the magnetic field configuration in the lobes and its observational consequences, using a realistic model for the magnetic field in the intracluster medium, very high density contrast in the lobes and high numerical resolution. We confirm, now with a realistic magnetic field model, that lobes have characteristic trajectories in the radio power/linear size diagram which depend strongly on their environment. We investigate the detailed evolution of polarized emission, showing that the lobes evolve from the initially ordered field configuration imposed by our boundary conditions to one in which the longitudinal field comes to dominate. We obtain simulated observations of polarization whose properties are quantitatively consistent with observations. The highly spatially intermittent magnetic field also reproduces the observation that inverse-Compton emission from lobes is much smoother than synchrotron. Our simulations allow us to study the depolarizing effect of the external medium on the lobes, and so to demonstrate that Faraday depolarization from environments of the type we consider can reproduce the integrated fractional polarization properties of large samples and the observed preferential depolarization of the receding lobe.