Abstract The current state of the art in pulsar magnetosphere modelling assumes the force-free limit of magnetospheric plasma. This limit retains only partial information about plasma velocity and neglects plasma inertia and temperature. We carried out time-dependent 3D relativistic magnetohydrodynamic (MHD) simulations of oblique pulsar magnetospheres that improve upon force free by retaining the full plasma velocity information and capturing plasma heating in strong current layers. We find rather low levels of magnetospheric dissipation, with < 10 per cent of pulsar spin-down energy dissipated within a few light cylinder radii, and the MHD spin-down that is consistent with that in force free. While oblique magnetospheres are qualitatively similar to the rotating split-monopole force-free solution at large radii, we find substantial quantitative differences with the split-monopole, e.g., the luminosity of the pulsar wind is more equatorially concentrated than the split-monopole at high obliquities, and the flow velocity is modified by the emergence of reconnection flow directed into the current sheet.