We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric dynamical systems. These systems are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. We show that, for some families of reaction networks, this new class is much larger than the class of toric systems. For example, for some networks we may even go from an empty locus of toric systems in parameter space to a positive-measure locus of disguised toric systems. We focus on the characterization of the disguised toric locus by means of (real) algebraic geometry.