In inertial confinement fusion (ICF) experiments, a capsule filled with fuel is imploded by shock waves generated by laser-driven ablation. The goal is to ignite a fusion burn at the centre of the implosion, and have that burn propagate through and consume the inertially confined fuel. Hydrodynamic instabilities cause mixing between the capsule material and the fuel, which is highly detrimental to the propagation of a fusion burn. A key hydrodynamic instability in ICF is the Richtmyer-Meshkov instability (RMI), which occurs when a shock interacts with a perturbed density interface. In ICF and astrophysical applications, the RMI typically occurs in plasmas. Here, we study the RMI of thermal interfaces in the context of the ideal two-fluid, ion-electron, continuum equations. These couple a separate set of conservation equations for each species to the full Maxwell equations. To clearly elucidate the impact of two fluid effects, we simulate cases with Debye lengths ranging from a hundredth to a tenth of the interface perturbation wavelength. These simulations reveal a wealth of new physical phenomena, implying that the RMI may be more detrimental to ICF than predicted by single-fluid models: interfacial growth rates are substantially greater, electrically driven interface oscillations drive variable acceleration Rayleigh-Taylor instability, and shocks become distorted, which may affect their convergence. The possibility of suppressing the RMI with a seed magnetic field will also be explored in the two-fluid plasma model.