This paper presents a new robust distributed approach for the energy management of rural multi-microgrids communities in Chile. These rural communities are often geographically isolated and have a poor connection to the main electricity grid. Operating them as individual microgrids allows the community to connect to the grid or behave as a stand-alone system (islanded). Connecting and coordinating the operation of the microgrids can improve the system's robustness to energy and demand uncertainty, as well as reducing the dependency on the main grid. However, this coordination can be challenging because the communication between the different microgrids can sometimes deteriorate, moreover, both load and local generation profiles are challenging to predict. For these reasons, in this paper, a new robust distributed hierarchical Energy Management System (EMS) for coordination of multiple microgrids (multi-microgrids) is introduced, which uses robust distributed Model Predictive Control (MPC). Distributed optimisation permits improving reliability in the presence of poor or lack of communications between microgrids. To consider the uncertainty associated with renewable energy generation and load consumption, fuzzy prediction interval models are included in the MPC design. The proposed method is compared with robust centralised MPC and deterministic distributed MPC to assess its performance. Simulation results are presented using data representative of a rural community in coastal Chile applied to the robust MPC design. The outcomes of this work demonstrate that the proposed robust distributed EMS can improve the power supply, apply peak shaving, and can successfully operate when there is a loss of communication between microgrids. The use of decreasing fuzzy intervals allows an accurate prediction of both load and renewable energy generation, making the system robust to uncertainty in the available resources. Moreover, results show that in absence of communication, the power supply obtained under different scenarios with the distributed EMS is better than the equivalent centralised EMS.