The main point we address in this paper is the question of thermodynamic stability for phase-separating systems, at coexistence in equilibrium. It has long been known that numerical simulations of different statistical models may yield "Van der Waals-like" isotherms in the coexistence region. Such "inverted" convexity segments of thermodynamic fields, known as unstable, are forbidden by the second law of thermodynamics on entropy, and their presence is not justified in exact results. In numerical experiments, their origin has been associated with the interface between the two coexisting phases. Nonetheless, the violation of the second law by entropy has not yet, to our knowledge, been rationalized. In this work, we introduce the thermodynamics of the interface between coexisting phases and give an alternative interpretation to the theory developed by Hill in the 1960s. Our approach points to a misinterpretation of the usual measurements of thermodynamic potentials in simulations. Correct interpretation eliminates the unstable regions of the true potentials. Our adapted theory is verified for the 2D lattice gas through carefully planned simulations. The thermodynamic description of the interface behavior inside the coexistence region restores the proper convexity of the true chemical potential isotherms. As a bonus, our interpretation allows direct calculation of surface tension in very good accordance with Onsager's analytic prediction.