The phase behavior of a solution containing a mixture of large and small, cross-bridging ("two-sided sticker") particles is studied using a lattice model analyzed with the aid of mean-field calculations and Monte Carlo simulations. Neither the large nor the small particles interact with each other (except for excluded volume effects). However, the small particles can adsorb onto the surface of one or two large particles, in the latter case providing a cross-bridge, i.e., an adhesive bond, between the large particles. The formulation of the model is motivated by experimental studies involving aqueous solutions of vesicles (the large particles) and biotin-avidin-biotin cross-bridges. This system exhibits a first-order phase transition from a dilute to a condensed phase of vesicles once the average number of stickers per vesicle exceeds a certain threshold value. The statistical thermodynamic description of the system becomes particularly simple upon (Legendre) transformation from the two-component canonical ensemble to a "mixed" ensemble involving a constant chemical potential of the cross-bridge particles. The phase separation behavior of the system is calculated for two sets of molecular parametrs, revealing good qualitative agreement with relevant experiments.