We introduce a new framework for the numerical solution of hyperbolic systems of partial differential equations, based on two different formulations of the same governing equations: a conservative formulation and a primitive (nonconservative) one. While they are mathematically equivalent under smoothness assumptions, the latter becomes ill-posed in the presence of discontinuities due to nonconservative products. This fundamental issue extends to the discrete level; in fact, nonconservative schemes tend to converge to incorrect solutions. Multiple theories have been proposed to handle nonconservative products at both the analytical and discrete levels, but none has proven to be entirely successful. On the other hand, in several applications, primitive formulations offer significant advantages. For this reason, there has recently been growing interest in approaches that carefully exploit primitive-variable structures. Our framework is designed to combine the advantages of both formulations and can be effectively applied in several contexts, e.g., adaptivity, multifluid flows, and asymptotic-preserving schemes.