Shock-driven hydrodynamic instabilities in a plasma usually lead to interfacial mixing and the generation of electromagnetic fields, which are nonequilibrium processes coupling kinetics with meso- and macroscopic dynamics. The understanding and modeling of these physical processes are very challenging tasks for single-fluid hydrodynamic codes. This work presents a new framework that incorporates both kinetics and hydrodynamics to simulate shock waves and hydrodynamic instabilities in high-density plasmas. In this hybrid code, ions are modeled using the standard particle-in-cell method together with a Monte Carlo description of collisions while electrons are modeled as a massless fluid, with the electron heat flux and fluid–particle energy exchange being considered in the electron pressure equation. In high-density plasmas, Maxwell’s equations are solved using Ohm’s law instead of Ampère’s law. This hybrid algorithm retains ion kinetic effects and their consequences for plasma interpenetration, shock wave propagation, and hydrodynamic instability. Furthermore, we investigate the shock-induced (or gravity-induced) turbulent mixing between a light and a heavy plasma, where hydrodynamic instabilities are initiated by a shock wave (or gravity). This study reveals that self-generated electromagnetic fields play a role in the formation of baroclinic vorticity along the interface and in late-time mixing of the plasmas. Our results confirm the ability of the proposed method to describe shock-driven hydrodynamic instabilities in a plasma, in particular, nonequilibrium processes that involve mixing and electromagnetic fields at the interface.