We have investigated the phase behavior of a symmetrical binary fluid mixture for the situation where the chemical potentials μ1 and μ2 of the two species differ. Attention is focused on the set of interparticle interaction strengths for which, when μ1 = μ2, the phase diagram exhibits both a liquid-vapor critical point and a tricritical point. The corresponding phase behavior for the case μ1 μ2 is investigated via integral-equation theory calculations within the mean spherical approximation and grand canonical Monte Carlo (GCMC) simulations. We find that two possible subtypes of phase behavior can occur, these being distinguished by the relationship between the triple lines in the full phase diagram in the space of temperature, density, and concentration. We present the detailed form of the phase diagram for both subtypes and compare with the results from GCMC simulations, finding good overall agreement. The scenario via which one subtype evolves into the other is also studied, revealing interesting features.