With the widespread adoption of combined heat and power and power-to-heat technologies, multi-energy microgrids (MEMGs) have been garnering significant research attention from both industry and academia. However, dealing with uncertainties from renewable energy and load and coordinating multiple energy carriers are the main challenges for MEMG operation. In this regard, a two-stage robust operation method of electricity-gas-heat integrated MEMGs considering heterogeneous uncertainties is proposed in this paper. First, network models for an electricity-gas-heat-based distribution-level MEMG are formulated considering the dynamic characteristics of gas and heat networks. Then, the power-to‑hydrogen-and-heat unit and ladder-type carbon trading mechanism are introduced to reduce the curtailment of wind power and carbon emissions. Further, a two-stage robust optimization (TSRO) method is applied to tackle uncertainties of wind power and load under extreme scenarios in the MEMG operation by iteratively solving the operation problem with the column and constraint generation (C&CG) algorithm. Finally, case studies are conducted to verify our proposed method, demonstrating that it can reduce the multi-energy supply cost while the stepped carbon trading mechanism can also significantly reduce carbon emissions.