  We present a theory of spatially indirect exciton condensate states in
systems composed of a pair of electrically isolated Bernal graphene bilayers.
The ground state phase diagram in a two-dimensional
displacement-field/inter-bilayer-bias space includes layer-polarized
semiconductors, spin-density-wave states, exciton condensates, and states with
mixed excitonic and spin order. We find that two different condensate states,
distinguished by a chirality index, are stable under different electrical
control conditions.
