We derive the optimal signaling for a multiple input multiple output (MIMO) full-duplex two-way channel under the imperfect transmit front-end chain. We characterize the two-way rates of the channel by using a game-theoretical approach, where we focus on the Pareto boundary of the achievable rate region and Nash equilibia (NE). For a MISO full-duplex two-way channel, we prove that beamforming is an optimal transmission strategy which can achieve any point on the Pareto boundary. Furthermore, we present a closed-form expression for the optimal beamforming weights. In our numerical examples we quantify gains in the achievable rates of the proposed beamforming over the zero-forcing beamforming. For a general MIMO full-duplex channel, we establish the existence of NE and present a condition for the uniqueness of NE. We then propose an iterative water-filling algorithm which is capable of reaching NE. Through simulations the threshold of the self-interference level is found, below which the full-duplex NE outperforms the half-duplex TDMA.