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This paper proposes a multi-level multi-energy management framework for the coordinated and interactive operation of heterogeneous multi-microgrids (MMGs) based on many-criteria optimality. With the proposed framework, the highly nonlinear and complex MMG multi-energy management (MMGMEM) problem is formulated into tri-level scheduling subproblems with multi-energy couplings and multi-level interactions, in which the multi-energy trading with energy networks and multi-energy couplings within MGs are optimized in the upper and middle level, and a middle level is added to correct scheduling decisions of the upper level for coordinating the MMG multi-energy sharing. Then, a multi-step matrix decomposition technique is developed to decompose the high dimensional multi-energy coupling matrix of MMGs into the sum of three linear and sparse submatrices for improving the computation efficiency and scalability. Furthermore, a many-criteria decision making (MCDM) model is proposed for the multi-energy sharing problem to achieve an optimum tradeoff in which all microgrids (MGs) can benefit from electricity-gas exchanges, and an evolutionary many-objective optimization based on hyperplane transformation algorithm is used to solve the MCDM problem. Simulation results verify that the proposed framework can achieve a cost saving for each MG (over 19%), and validate its scalability in solving large-scale MMGMEM problems. • A tri-level interactive framework is proposed for the coordinated operation of MMGs. • An extended multi-energy coupling matrix with MMG interactions is formulated. • A multi-step matrix decomposition technique is used to reduce the problem complexity. • A many-criteria optimality model is proposed for the tradeoff of all MGs' benefits. • The utilization of renewables can be enhanced with the minimal total operation cost. [ABSTRACT FROM AUTHOR]