The onset of liquid-vapor separation in an interaction site model of a lysozyme aqueous solution is investigated by means of molecular dynamics (MD). Calculations are performed for a soft-core version of a potential early introduced by Carlsson et al. (J. Phys. Chem. B 2001, 105, 9040; 2001, 105, 12189.) whose liquid-vapor coexistence was studied by Rosch and Errington (J. Phys. Chem. B 2007, 111, 12591.); our modified model preserves the tailoring onto the experimental lysozyme solution properties embodied by those descriptions. We first show that the structural results obtained by Carlsson et al. at ambient conditions are quite well reproduced by our approach. Then, we cool the system along an isochoric path by monitoring the structural and dynamical properties at various temperatures. We thus show that a fluid-fluid separation takes place at a temperature 15% below the presumed binodal; in particular, we observe that density inhomogeneities develop rather early in the MD run and evolve over tens of nanoseconds into two dense aggregates that eventually merge, after similar to 100 ns more, into a single liquid phase separated from a vapor region by a well-defined planar interface. The densities of the two coexisting fluids are compatible with previous determinations of the binodal line. The connections of this work to the overall scenario of phase stability investigations in protein solutions, as well as possible developments based on the use of more refined models, are discussed.