This paper investigates the residential energy management problems in the power-heat-coupling system. For better solving this issue, based on the novel concept of We-Energy (WE) for energy Internet (EI), this paper proposes a residential WE (R-WE) framework faced by the terminal users who play an important role in renewable energy consumption. Inspired by the full-duplex feature of R-WE, the power and heat supply-demand balance constraints are fulfilled in a regional unit of EI, but it may be inequality constraints for corresponding consumer, producer, or WE operation modes in only one R-WE. The R-WE framework can be classified into four operation modes, which is island mode, consumer mode, producer mode, and WE mode. On this basis, the R-WE models cooperate to achieve the objective of minimizing the operation costs, smoothing out the loads’ variations, and renewable resource fluctuations. Specifically, the R-WE framework with respect to the heat-power-coupling problem can be solved in the distributed double-consensus algorithm (DDCA), which is designed two sets of consensus algorithms by using four completely different consensus variables to calculate the power and heat multipliers, evaluate the power and heat generations and, thereby, further obtain the power mismatches of electricity and heat. Also, the Karush-Kuhn-Tucker (KKT) optimal conditions of the proposed DDCA is further proved. Meanwhile, a novel projection operation method for combined heat and power devices is designed to take the infeasible solutions mapped into the feasible region. Finally, the simulation results in different cases further demonstrate that the proposed R-WE frame can be an appropriate method to analyze the terminal consumers and harmonize multienergy producers in the process of constructing EI projects. And an R-WE framework of the campus has experimented with minority loads in island mode.