The promotion of carbon neutrality targets will significantly influence the energy management of multi micro energy grid (MMEG) systems that can supply flexible local loads using various distributed energy resources. So far, few works incorporate the carbon emissions management with the fully-distributed coordination of MMEG systems to protect the environment and individual privacy and improve the energy efficiency. To fill this gap, in this paper, a hierarchical optimization framework is proposed for the MMEG system to solve an environmental economic dispatch problem, such that the energy coordination strategy could be obtained in a fully-distributed way. At the upper level, under a local and time-varying communication network, we develop an aggregate model based on a real-time price policy for the MMEG system. On this basis, a fully distributed consensus-based gradient projection method is proposed to minimize the total system generation cost with the coupled emission constraint satisfied, wherein a piecewise linearization technique is applied to handle the non-convexity and non-smoothness of the cost function caused by valve-point loading effects (VPLE) of thermal units. At the lower level, we apply an accelerated distributed augmented Lagrangian (ADAL) method to deal with a cost minimization optimization problem for users, such that each user could implement its own optimal strategy simultaneously based on the obtained aggregated strategy from the upper level. The effectiveness of the proposed approach is demonstrated by numerical simulations on an MMEG test system.