The thermal transport of magnons has attracted substantial attention as an energy-efficient alternative to the transport of electrons. Most theoretical studies so far have been carried out within the frame of the linear spin-wave theory, which dramatically fails upon increasing the temperature and in the presence of competing interactions. In this work, we consider the impact of three- and four-magnon interactions in a honeycomb antiferromagnet, where such interactions are remarkably strong even at zero temperature. Using a combination of quantum field theory and mean-field theory, we compute the band structure of the interacting magnons and investigate the spin Nernst effect. We find that in the presence of in-plane Dzyaloshinskii-Moriya Interaction, the three-magnon interaction induces a non-reciprocal band splitting, even at zero temperature, leading to an enhancement of the spin Nernst conductivity. In contrast, the four-magnon interaction renormalizes the magnon spectrum at high temperatures, leading to a reduction of the overall magnon spin Nernst effect. These results suggest that interactions can massively influence the transport properties of magnons in antiferromagnets, even at zero temperature, and should be taken into account for predictive modeling.

15 pages, 8 figures