  We present a stability analysis on a driven-dissipative electron-hole
condensate in the BCS (Bardeen-Cooper-Schrieffer)-BEC
(Bose-Einstein-condensation)-crossover region. Extending the combined
BCS-Leggett theory with the generalized random phase approximation (GRPA) to
the non-equilibrium case by employing the Keldysh formalism, we show that the
pumping-and-decay of carriers causes a depairing effect on excitons. This
phenomenon gives rise to an attractive interaction between excitons in the BEC
regime, as well as a supercurrent that anomalously flows anti-parallel to
$\nabla \theta({\bf r})$ (where $\theta({\bf r})$ is the phase of the
condensate) in the BCS regime, both leading to dynamical instabilities of an
exciton-BEC. Our result suggests that substantial region of the exciton-BEC
phase in the phase diagram (in terms of the interaction strength and the decay
rate) is unstable.
