This paper proposes a novel Bayesian reciprocity calibration method that consistently ensures uplink and downlink channel reciprocity in repeater-assisted multiple-input multiple-output (MIMO) systems. The proposed algorithm is formulated under the minimum mean-square error (MMSE) criterion. Its Bayesian framework incorporates complete statistical knowledge of the signal model, noise, and prior distributions, enabling a coherent design that achieves both low computational complexity and high calibration accuracy. To further enhance phase alignment accuracy, which is critical for calibration tasks, we develop a von Mises denoiser that exploits the fact that the target parameters lie on the circle in the complex plane. Simulation results demonstrate that the proposed MMSE algorithm achieves substantially improved estimation accuracy compared with conventional deterministic non-linear least-squares (NLS) methods, while maintaining comparable computational complexity. Furthermore, the proposed method exhibits remarkably fast convergence, making it well suited for practical implementation.