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Many sensor fusion approaches based on the Kalman filter or its variants assume that sensor measurements are disturbed by a white Gaussian noise, which implies an observation error statistically independent of the state estimate. These methods are often being applied in situations where the white noise assumption may not be satisfied, which potentially leads to overconfidence and a divergence of the filter. In this paper, we derive a new Kalman gain formula that provides an optimal update rule in the presence of a known correlation between errors in the state estimate and an observation, which is caused by a presence of a shared error term. The new method is described in the context of the Ensemble Kalman filter, where such a correlation can be directly estimated from the state and observation samples. The proposed generalised Ensemble Kalman filter is evaluated in a scenario where a mobile robot estimates its global position by fusing visual odometry data with an auto-correlated sequence of measurements from a stand-alone Global Positioning System (GPS) receiver.