In microgrid (MG) systems, traditional centralised energy trading models can lead to issues such as low energy efficiency due to unstable energy supply and lack of flexibility. Peer-to-peer (P2P) trading models have been widely used due to their advantages in promoting the sustainable development of renewable energy and reducing energy trading costs. However, P2P multi-energy trading requires mutual agreements between two microgrids (MGs), and the uncertainties of renewable energy and load affects energy supply security. To address these issues, this article proposed a distributed robust operation strategy based on P2P multi-energy trading for multi-microgrid (MMG) systems. Firstly, a two-stage robust optimisation (TRO) method was adopted to consider the uncertainties of P2P multi-energy trading between MGs, which reduced the conservatism of robust optimisation (RO). Secondly, a TRO model for P2P multi-energy trading among MGs was established based on the Nash bargaining theory, where each MG negotiates with others based on their energy contributions in the cooperation. Additionally, a distributed algorithm was used to protect the privacy of each MG. Finally, the simulation results based on three MGs showed that the proposed approach can achieve a fair distribution of cooperative interests and effectively promote cooperation among MGs.