The stability of a planar, two-component, charged lipid bilayer with respect to lateral phase separation is analysed. Particular focus is put on the influence of electrostatic coupling between the two apposed monolayers. We describe the membrane on the mean-field level by a lattice gas model, supplemented by the Poisson-Boltzmann theory for electrostatic interactions. The spinodal line is calculated analytically for a symmetric membrane. We demonstrate that the critical non-ideality parameter decreases with the extent of electrostatic coupling across the bilayer. In the limit of strong coupling it approaches the value of an uncharged membrane.