The problem of synthesizing reactive mass-exchange networks (REAMENs) is addressed for which the equilibrium relations governing the transfer of a certain species from rich streams to reactive mass-separating agents are nonlinear (convex and/or nonconvex). Previously developed synthesis procedures for REAMENs are not applicable to such cases, since they were developed for cases of linear and/or convex equilibria. Two peculiar phenomena are associated with REAMENs with general nonlinear equilibria : the location of a mass-exchange pinch point is not restricted to supply compositions of streams, mass exchangers may straddle the pinch point even for networks featuring the minimum cost of mass-separating agents. To account for such intriguing characteristics, a systematic procedure is developed to synthesize cost-effective REAMENs with general nonlinear equilibria. In addition, a robust solution procedure is devised to guarantee the identification of the global solution of the developed optimization programs. Applicability and usefulness of the proposed synthesis procedure are demonstrated via a case study on the desulfurization of gaseous wastes from a coal-to-methanol plant.