Condensate formation of biopolymer solutions, prominently those of various intrinsically disordered proteins (IDPs), is often driven by "sticky" interactions between associating residues, multivalently present along the polymer backbone. Using a ternary mean-field "stickers-and-spacers" model, we demonstrate that if sticker association is of the order of a few times the thermal energy, a delicate balance between specific binding and nonspecific polymer-solvent interactions gives rise to a particularly rich ternary phase behavior under physiological circumstances. For a generic system represented by a solution comprising multiassociative scaffold and client polymers, the difference in solvent compatibility between the polymers modulates the nature of isothermal liquid-liquid phase separation (LLPS) between associative and segregative. The calculations reveal regimes of dualistic phase behavior, where both types of LLPS occur within the same phase diagram, either associated with the presence of multiple miscibility gaps or a flip in the slope of the tie-lines belonging to a single coexistence region.