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This paper proposes a reconfigurable intelligent surface-aided two-way full-duplex communication with non-orthogonal multiple access transmission schemes to improve spectrum utilization. Besides, the joint impact of error phase-shift quantization, imperfect successive interference cancellation, and residual loop interference on the system performance have also been investigated. Under Nakagami-m fading channels, approximate closed-form expressions for the outage performance and the ergodic rate are derived. Through asymptotic analyses, some insights are achieved, including the diversity order, the coding gain, and the ergodic slope. Moreover, three adaptive power allocation optimization problems are formulated aiming to: 1) minimize outage probability, 2) achieve max-min rate fairness, and 3) maximize the user’s sum rate subject to the quality-of-service constraint. Simulation results not only validate …