Radiative thermal diodes based on two-element structures rectify heat flows thanks to a temperature dependence of material optical properties. The heat transport asymmetry through these systems, however, remains weak without a significant change in material properties with the temperature. Here we explore the heat transport in three-element radiative systems and demonstrate that a strong asymmetry in the thermal conductance can appear because of many-body interactions, without any dependence of optical properties on the temperature. The analysis of transport in three-body systems made with polar dielectrics and metallic layers reveals that rectification coefficients exceeding 50% can be achieved in the near-field regime with temperature differences of about 200 K. This work paves the way for compact devices to rectify near-field radiative heat fluxes over a broad temperature range and could have important applications in the domain of nanoscale thermal management.