The phase diagram of a system constituted of neutrons, protons, $\Lambda$-hyperons and electrons is evaluated in the mean-field approximation in the complete three-dimensional space given by the baryon, lepton and strange charge. It is shown that the phase diagram at sub-saturation densities is strongly affected by the electromagnetic interaction, while it is almost independent of the electric charge at supra-saturation density. As a consequence, stellar matter under the condition of strangeness equilibrium is expected to experience a first as well as a second-order strangeness-driven phase transition at high density, while the liquid-gas phase transition is expected to be quenched. An RPA calculation indicates that the presence of this critical point might have sizable implications for the neutrino propagation in core-collapse supernovae.