This paper proposes a Nash bargaining cooperative game model for a microgrid cluster system with double re-energy-load delay considering electricity, heat and gas multi-energy synergies. With the minimization of the operating cost of each microgrid as the objective function, a low-carbon operation model of a multi-energy complementary integrated energy microgrid considering fuzzy opportunity constraints is developed, an optimal operation mechanism including carbon quota and carbon trading is assessed, and a carbon capture system and an electricity-gas conversion device are added to the improved cogeneration unit model. Source-load uncertainty in microgrids is described in terms of new fuzzy parameters of new energy and undefined parameters of load demand. Each microgrid plays a second game with the marginal contribution rate and carbon trading cost rate as the bargaining power. The model is solved in a distributed manner using the ADMM-RGE algorithm. Finally, the simulation results show that the proposed multi-microgrid power-sharing way maximizes the benefits of microgrid alliances; the cooperative help of microgrid alliances is pretty distributed according to the size of each microgrid's energy contribution; carbon capture joint power-gas systems and energy sharing methods between microgrids can effectively reduce carbon emissions during microgrid operation.