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This article presents a new MSE-based transceiver design for the MIMO interference channel (IC). The proposed design monotonically decreases the maximum per-stream MSE of all users by alternatively optimizing between a minimum MSE solution for the equalizers and a second-order-cone-programming (SOCP) solution for the precoders. As the system's error rate performance at high signal-to-noise ratio (SNR) is dominated by the spatial stream with the largest MSE, minimizing the maximum per-stream MSE not only improves the overall system performance but also ensures the fairness in error rate among users. The performance advantage of the proposed algorithm is verified by the numerical simulations.