Summary: We consider the flow of an electrically conducting fluid confined in a rotating spherical shell. The flow is driven by a directly imposed electromagnetic body force, created by the combination of an electric current flowing from the inner sphere to a ring-shaped electrode around the equator of the outer sphere and a separately imposed predominantly axial magnetic field. We begin by numerically computing the axisymmetric basic states, which consist of a strong zonal flow. We next compute the linear onset of non-axisymmetric instabilities, and fully three-dimensional solutions up to ten times supercritical. We demonstrate that an experimental liquid-sodium device 50 cm in diameter could achieve and exceed these parameter values.