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Theoretical studies have shown that Majorana bound states can be induced at the ends of a one dimensional wire, a phenomenon possible due to the interplay between s-wave superconductivity, spin-orbit coupling, and an external magnetic field. These states have been observed in superconductor-semiconductor hybrid nanostructures in the presence of a Zeeman field, and in the limit of a low density of particles. In this paper, we demonstrate and discuss the possibility of the emergence of Majorana bound states in a superconducting Rashba nanowire deposited on an antiferromagnetically ordered surface. We calculate the relevant topological invariant in several complementary ways. Studying the topological phase diagram reveals two branches of the non trivial topological phase -- a main branch, which is typical for Rashba nanowires, and an additional branch emerging due to the antiferromagnetic order. In the case of the additional topological branch, Majorana bound states can also exist close to half-filling, obviating the need for either doping or gating the nanowire to reach the low density regime. Moreover, we show the emergence of the Majorana bound states in the absence of the external magnetic field, which is possible due to the antiferromagnetic order. We also discuss the properties of the bound states in the context of real space localization and the spectral function of the system. This allows one to perceive the band inversion within the spin and sublattice subspaces in the additional branch, contrary to the main branch, where the only band inversion reported in previous studies exists in the spin subspace. Finally, we demonstrate how these topological phases can be confirmed experimentally in transport measurements.