Bilayer electron-hole systems with unequal electron and hole densities are expected to have exciton condensate ground states with spontaneous spin-polarization in both conduction and valence bands. In the absence of spin-orbit and electron-hole exchange interactions there is no coupling between the spin-orientations in the two quantum wells. In this article we show that Rashba spin-orbit interactions lead to unconventional magnetic anisotropies, whose strength we estimate, and to ordered states with unusual quasiparticle spectra.