We study the existence of thermodynamic instabilities in the nuclear equation of state relative to the high density regime reached in the central core of compact stars. In the framework of a relativistic mean-field theory, we analyze the asymmetric nuclear properties in beta-equilibrium, including hyperons and Delta-isobar degrees of freedom. We investigate a finite density phase transition characterized by pure hadronic matter with the presence of mechanical instability (relative to the fluctuation of baryon number) and of chemical-diffusive instability (relative to the fluctuation of electric charge concentration). We find that, in the presence of thermodynamic instabilities, two hadronic phases with different values of electric charge content may coexist, with several phenomenological consequences in the physics of compact stars.