The ground-state energy of strongly coupled binary ionic mixtures is evaluated by solving the hypernetted-chain (HNC) equations for the pair distribution functions. The point ions are immersed in a rigid and uniform neutralizing background of relativistic electrons. We have calculated the equation of state and the pair structure of these binary fluids for three concentrations and over a rather extensive range of densities. The miscibility essentially depends on the statistics of the species and on the coupling. We show that the phase separation is very sensitive to the choice of the wave function used to describe the mixture.