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We use optical reflectometry and surface pressure techniques to measure co-adsorption of the anionic surfactant sodium dodecyl sulfate (SDS) and the protein lysozyme at the air-aqueous interface. We observe lysozyme/SDS co-adsorption behavior in two different buffers for which solution-phase binding data are available in the literature. The co-adsorption of lysozyme/SDS complexes is controlled by the mode of protein/surfactant binding that occurs in solution. In a pH 5.0 acetate buffer, the extent of co-adsorption is weakly dependent on SDS concentration throughout the specific and transitional binding regimes. In a pH 6.9 phosphate buffer, the extent of co-adsorption is weakly dependent on SDS concentration in the specific binding regime, but it increases dramatically, giving rise to multilayer co-adsorption, in the transitional binding regime. In both buffers, the extent of co-adsorption dramatically decreases in the cooperative binding regime. Lysozyme/SDS co-adsorption is strongly influenced by kinetically trapped non-equilibrium adsorbed layer states, such that adsorbed amounts are markedly path-dependent. Surface pressure measurements by themselves do not capture the variations in adsorption in the different binding regimes, nor do they capture the path-dependency of co-adsorption.