Relay selection strategies help to improve spectral and energy efficiencies, to enhance transmission robustness, or to reduce latency in cooperative networks. A novel relay selection strategy, which maximizes the overall achievable sum rate, is proposed and analyzed for multiple-input multiple-output (MIMO) amplify-and-forward two-way relay networks with spatial multiplexing. Interestingly, this strategy amounts to maximizing the minimum of the determinant of the Wishart matrices from the relays to the two user nodes. The performance of this strategy is investigated by deriving the average sum rate approximations in closed-form. Our analysis and numerical results reveal that the proposed relay selection scheme provides substantial sum rate improvements.