To fully explore the potential value of the integrated energy microgrid (IEM) and promote the consumption of renewable energy, a two-layer game-based distribution network-multiple IEM cooperative optimization model is proposed to address the problem of cooperative management of multiple IEMs in the same distribution network. For the construction of the IEM model, a carbon capture system and an electricity-to-gas conversion device are added to the cogeneration units to obtain low-carbon benefits. At the same time, robust interval planning is used to deal with the problems of renewable energy and load uncertainty in the IEM. This paper initially establishes the framework of the alliance system between the DSO and IEMs, analyzing the game dynamics among various players. Subsequently, the two-layer game is further categorized into a master-slave game and a cooperative game. The DSO, as the game leader, sets the tariff to guide the IEM coalition to optimize with the goal of maximizing its own benefit. And the IEM coalition, as the game follower, responds to the DSO's decision by cooperating with members to share energy with the goal of minimizing its own operating cost. At the same time, Nash negotiation theory is used to solve the cooperative operation problem of the IEM coalition, and the dichotomous method and alternating direction multiplier method (ADMM) are combined to analyze the model. Finally, an illustrative example is presented to validate the feasibility and effectiveness of the proposed model and methodology.