We have developed a formalism which accounts for the kinetics of precipitation of binary non-ideal solid-solutions, and includes nucleation, growth and/or re-dissolution processes. It allows prediction of the time evolution of the aqueous ion activities together with all characteristics of the solid phase: crystal size distribution function and composition profiles of particles. We show that different scenarios of precipitation are possible depending on the thermodynamic properties of the solid solution end-members, the degree of non-ideality of the solid solution and the initial conditions of precipitation. We illustrate the various scenarios by simulations performed under a few simplifying assumptions for some carbonate solid solutions of geochemical interest. To our knowledge, this is the first time in the field of geochemistry that these out-of-equilibrium precipitation processes of non-ideal solid solutions are fully described.