  We study the influence of quantizing perpendicular magnetic fields on the
ground state of a bilayer with electron and hole fluids separated by an opaque
tunnel barrier. In the absence of a field, the ground state at low carrier
densities is a condensate of s-wave excitons that has spontaneous interlayer
phase coherence. We find that a series of phase transitions emerge at strong
perpendicular fields between condensed states and incompressible incoherent
states with full electron and hole Landau levels. When the electron and hole
densities are unequal, condensation can occur in higher angular momentum
electron-hole pair states and, at weak fields, break rotational symmetry. We
explain how this physics is expressed in dual-gate phase diagrams, and predict
transport and capacitively-probed thermodynamic signatures that distinguish
different states.
