The mutual optimization of a multi-microgrid integrated energy system (MMIES) can effectively improve the overall economic and environmental benefits, contributing to sustainability. Targeting a scenario in which an MMIES is connected to the same node, an energy storage coordination control strategy and carbon emissions management strategy are proposed, and an adaptive step-size method is applied to improve the distributed optimization of MMIESs based on the alternating direction multiplier method (ADMM). Firstly, the basic framework of MMIESs is established, and a coordinated control strategy limiting the time of charge and the discharge of the battery storage system (BSS) is proposed. Then a multi-objective optimization model based on operating and environmental cost is formulated. Considering that different microgrids may be managed by different operators and a different convergence speed of multi-objective optimization iteration, an adaptive step-size distributed iterative optimization method based on ADMM is used, which can effectively reduce the cost and protect the privacy of each microgrid. Finally, a system composed of three microgrids is taken as an example for simulation analysis. The results of distributed optimization are accurate, and the proposed coordinated control strategy can effectively enhance the revenue of ESS, which verifies the effectiveness of the proposed method.