We explore the physics of relativistic radiation mediated shocks (RRMSs) in the regime where photon advection dominates over photon generation. For this purpose, a novel iterative method for deriving a self-consistent steady-state structure of RRMS is developed, based on a Monte-Carlo code that solves the transfer of photons subject to Compton scattering and pair production/annihilation. Systematic study is performed by imposing various upstream conditions which are characterized by the following three parameters: the photon-to-baryon inertia ratio ξ u * , the photon-to-baryon number ratioñ, and the shock Lorentz factor γ u . We find that the properties of RRMSs vary considerably with these parameters. In particular, while a smooth decline in the velocity, accompanied by a gradual temperature increase is seen for ξ u * ≫ 1, an efficient bulk Comptonization, that leads to a heating precursor, is found for ξ u * 1. As a consequence, although particle acceleration is highly inefficient in these shocks, a broad non-thermal spectrum is produced in the latter case. The generation of high energy photons through bulk Comptonization leads, in certain cases, to a copious production of pairs that provide the dominant opacity for Compton scattering. We also find that for certain upstream conditions a weak subshock appears within the flow. For a choice of parameters suitable to gamma-ray bursts, the radiation spectrum within the shock is found to be compatible with that of the prompt emission, suggesting that subphotospheric shocks may give rise to the observed non-thermal features despite the absence of accelerated particles.