Relativistic shocks have a central role in high energy astrophysical phenomena, with Gamma-Ray Bursts being the most prominent example. Their propagation in perfectly conductive plasmas has been extensively studied. In the present work we extend the analysis by assuming a finite electrical conductivity for the propagation medium and a finite thickness for the shock front. These two assumptions necessitate the inclusion of an additional jump condition derived through the covariant Gauss-Ampère Law and introduce a dimensionless parameter which depends on the electrical conductivity of the plasma in the shock front, the shock thickness, as well as on the shock's propagation four-velocity. We show that this parameter determines the degree to which the shock interacts with the propagation medium’s electromagnetic field and governs shock dynamics.