We present results from small-angle x-ray scattering data on the effect of high pressure on the phase behavior of dense lysozyme solutions in the liquid-liquid phase separation region, and characterize the underlying intermolecular protein-protein interactions as a function of temperature and pressure in this region of phase space. A reentrant liquid-liquid phase separation region has been discovered at elevated pressures, which originates in the pressure dependence of the solvent-mediated protein-protein interactions.