We study the phase behavior of mixtures of oppositely charged nanoparticles, both theoretically and experimentally. As an experimental model system we consider mixtures of lysozyme and lysozyme that has been chemically modified in such a way that its charge is nearly equal in magnitude but opposite in sign to that of unmodified lysozyme. We observe reversible macroscopic phase separation that is sensitive not only to protein concentration and ionic strength, but also to temperature. We introduce a heterogeneous Poisson-Boltzmann cell model that generally applies to mixtures of oppositely charged nanoparticles. To account for the phase behavior of our experimental model system, in addition to steric and electrostatic interactions, we need to include a temperature-dependent short-ranged interaction between the lysozyme molecules, the exact origin of which is unknown. The strength and temperature dependence of the short-ranged attraction is found to be of the same order of magnitude as that between unmodified lysozyme molecules. The presence of a rather strong short-ranged attraction in our model system precludes the formation of colloidal liquid phases (or complex coacervates) such as those typically found in mixtures of globular protein molecules and oppositely charged polyelectrolytes.