We present a detailed study of a recently reported discontinuity and bistability in a 12-nm GaAs/AlAs single-barrier p-i-n heterostructure, where a system of spatially separated two-dimensional electron and hole (e-h) layers of equal and tunable density is realized. Both features appear at TՇ300 mK and are strongly enhanced in a magnetic field Bտ10 T perpendicular to the layers, whereas they are suppressed by Bϳ1 T parallel to the layers. They correspond to a discontinuity in the e-h density and in the phase of the current magneto-oscillations. Whereas the high-current state has the expected properties of the uncoupled e-h layers, the low-current state behaves anomalously under all circumstances, and we identify them with a gas of spatially indirect excitons with binding energy 0.03 meVՇE b Շ0.3 meV and 0.5 meVՇE b Շ5 meV at B ϭ0 and Bϭ10 T, respectively. We interpret the bistability as a transition between the two regimes, which arises because of the competition between the in-plane screening, determined by the average e-e (h-h) distance and the magnetic length, and the interlayer e-h attraction.