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• Peer-to-peer energy trading problem of multi-energy microgrids are investigated. • The concept of energy trading consistency is firstly proposed. • The off-design performance model of the energy conversion device is considered. • The decision-making problem is solved by multi-agent soft actor-critic approach. Multi-energy microgrid technology is an essential for addressing the diversification of energy demand and local consumption of renewable energy sources. Peer-to-peer energy trading has emerged as a promising paradigm for the design of a decentralized trading framework. Therefore, this paper investigated the external peer-to-peer energy trading problem and internal energy conversion problem of interconnected multi-energy microgrids. The concept of energy trading consistency to avoid unreasonable energy trading behavior is first proposed and an off-design performance model of the energy conversion device is considered to more accurately reflect the operating status of the device. The complex decision-making problem with significantly large high-dimensional data is formulated as a partially observable Markov decision process and solved using the proposed multi-agent deep reinforcement learning approach combining the centralized training decentralized execution framework and soft actor-critic algorithm. Finally, the effectiveness of the proposed method was verified through a case simulation. The simulation results showed that the proposed method can reduce the total cost compared with the rule-based method. [ABSTRACT FROM AUTHOR]