  We study phase transitions in classical spin ice at nonzero magnetization, by
introducing a mean-field theory designed to capture the interplay between
confinement and topological constraints. The method is applied to a model of
spin ice in an applied magnetic field along the [100] crystallographic
direction and yields a phase diagram containing the Coulomb phase as well as a
set of magnetization plateaux. We argue that the lobe structure of the phase
diagram, strongly reminiscent of the Bose-Hubbard model, is generic to Coulomb
spin liquids.
