We consider a Boltzmann-like kinetic model describing a real gas with (possible) multiple inelastic collisions, including reactive collisions. We prove the global-in-time existence and uniqueness of solutions to the initial value problem associated to the model, in the case of collision-reaction laws compatible with a generalized form of the so-called Povzner inequality. The proof applies recent results on a class of nonlinear evolution equations in ordered Banach spaces, obtained by abstracting common order-monotonicity properties of some models of the so-called collisional kinetic theory.