We consider the phase separation in an asymmetrically charged lipid bilayer membrane consisting of neutral and negatively charged lipids that are in contact with in and out ionic solutions having different ionic strengths. The two asymmetric leaflets are coupled through electrostatic interactions. Based on a free-energy approach, the critical point and phase diagrams are calculated for different ionic strengths of the two solutions and coupling parameter. An increase of the coupling constant or asymmetry in the salt concentration between the in and out solutions yields a higher phase-separation temperature because of electrostatic interactions. As a consequence, the phase-coexistence region increases for strong screening (small Debye length). Finally, possible three-phase coexistence regions in the phase diagram are predicted.