  We study near-neighbour and dipolar Ising models on a lattice of
corner-sharing octahedra. In an extended parameter range of both models,
frustration between antiferromagnetism and a spin-ice-like three-in-three-out
rule stabilises a Coulomb phase with correlated dipolar and quadrupolar spin
textures, both yielding distinctive neutron-scattering signatures. Strong
further-neighbour perturbations cause the two components to order
independently, resulting in unusual multi-$k$ orders. We propose experimental
realisations of our model in rare-earth antiperovskites.
