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This study proposes a comprehensive framework for developing a multi-energy off-grid microgrid with the decoupled flow of thermal and electrical energies in a rural setting. A carbon-neutral microgrid with a hybrid generation system constituting a photovoltaic unit and a biofuel generator is proposed. In order to enhance the fuel utilisation efficiency, the biofuel generator is operated in combined cooling, heating, and power mode, and the recovered thermal energy forms the heat distribution network in the microgrid. The flexibility of system operation is improved by suitable multi-energy (electrical and thermal) storage. Firstly, an optimal sizing framework has been developed for the system as a mixed integer linear programming model. Secondly, a coordinated multi-energy management system (MEMS) has been developed for combined optimal dispatch of multiple generation and storage resources. The MEMS has been developed as a mixed integer non-linear programming model, which minimises system operational cost while considering minimum battery degradation to prolong its lifetime. Finally, a detailed economic analysis of the proposed system has been presented, highlighting the levellised cost of energy and net present value. Extensive case studies and simulation results depict the effectiveness and suitability of the proposed MEMS for the rural off-grid microgrid.