Mixtures of lysozyme and sodium dodecyl sulfate, SDS, are used to study protein - surfactant interactions. By precipitating the complex salt, CS(12), where DS- ions are the counterions to the positively charged lysozyme, a stoichiometric protein-surfactant component free from simple salts is synthesized. The solubility product of the complex salt is determined, and it provides a measure of the hydrophobic interaction in the precipitate. Aqueous systems of mixtures of the complex salt and the pure surfactant form a true three-component protein-surfactant-water system. The resulting phase behavior is studied and used as a fundament for theoretical discussion as well as modeling. With additional electrolyte excluded, the role of the electrostatic interactions is maximized, and the free energy balance between the different aggregates is shown to be determined by a balance between electrostatic and hydrophobic forces. We find three types of protein - surfactant aggregates: the insoluble complex salt, an electrostatically swollen gel-like state containing macroscopic aggregates, and a soluble complex containing a single protein molecule.