The phase behavior of protein solutions is affected by additives such as crowder molecules or salts. In particular, upon addition of multivalent counterions, a reentrant condensation can occur; i.e., protein solutions are stable for low and high multivalent ion concentrations but aggregating at intermediate salt concentrations. The addition of monovalent ions shifts the phase boundaries to higher multivalent ion concentrations. This effect is found to be reflected in the protein interactions, as accessed via small-angle X-ray scattering. Two simulation schemes (a Monte Carlo sampling of the counterion binding configurations using the detailed protein structure and an analytical coarse-grained binding model) reproduce the shifts of the experimental phase boundaries. The results support a consistent picture of the protein interactions responsible for the phase behavior. The repulsive Coulomb interaction is varied by the binding of multivalent counterions and additionally screened by any increase of the ionic strength. The attractive interaction is induced by the binding of multivalent ions, most likely due to ion bridging between protein molecules. The overall picture of these competing interactions provides interesting insight into possible mechanisms for tailoring interactions in solutions via salt effects.