The theoretical study of many-body effects in the context of near-field radiative heat transfer (NFRHT) has already led to the prediction of a plethora of thermal radiation phenomena. Special attention has been paid to nonreciprocal systems in which the lack of the Lorentz reciprocity has been shown to give rise to unique physical effects. However, most of the theoretical work in this regard has been carried out with the help of approaches that consider either pointlike particles or highly symmetric bodies (such as spheres), which are not easy to realize and explore experimentally. In this work we develop a many-body approach based on the thermal discrete dipole approximation (TDDA) that is able to describe the NFRHT between nonreciprocal objects of arbitrary size and shape. We illustrate the potential and the relevance of this approach with the analysis of two related phenomena, namely the existence of persistent thermal currents and the photon thermal Hall effect, in a system with several magneto-optical bodies. Our many-body TDDA approach paves the way for closing the gap between experiment and theory that is hindering the progress of the topic of NFRHT in many-body systems.