In the context of numerical simulation of turbulent compressible flows, many research efforts have been devoted to the design of methods discretely reproducing some fundamental physical properties that hold true in the continuous framework. These methods are often labeled as structure-preserving or physics-compatible and have been shown to be able to significantly enhance the fidelity and robustness of the numerical simulations. An important example is given by Pressure-Equilibrium Preserving (PEP) methods, in which the physical property of the compressible Euler equations to admit traveling density waves at constant velocity and pressure is discretely reproduced. In fact, the lack of discrete pressure equilibrium typically produces spurious pressure oscillations, drastically affecting the fidelity and stability of the numerical simulation. For the case of ideal gases, formulations that enforce both the PEP property and the conservation of primary invariants at a discrete level have been successfully developed, but when more complex Equations of State (EoS) are used, only approximated methods exist, unless the pressure evolution equation is discretized in place of the energy equation. In this last case, however, conservation of the total energy itself is sacrificed. Hybrid approaches that make use of the pressure evolution equation, in addition to the total-energy equation, have been tested for multicomponent flows, producing good results in terms of pressure oscillations and total-energy conservation. However, these approaches overspecify the pressure field, as the algebraic EoS is still retained for the advancement of the total-energy evolution equation. In this contribution, we investigate an approach in which the direct discretization of the pressure evolution equation is used in place of the algebraic equation of state, which guarantees exact total-energy conservation and pressure-equilibrium preservation at the cost of slightly perturbing the EoS. A numerical analysis is carried out to quantify the error between the various formulations, and numerical tests are presented to assess the achievement of the desired structural properties.