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This paper is concerned with distributed fusion (DF) estimation problem for nonlinear multi-sensor systems with correlated noises. Based on a recursive linear minimum variance estimation (RLMVE) framework, a novel filter is developed. It is proved that the RLMVE-based filter and the existing de-correlated filter have the functional equivalence. Then, for multi-sensor cases, cross-covariance matrices between any two local filters are derived. Based on the RLMVE-based filter and cross-covariance matrices, a DF filter weighted by matrices is proposed in the sense of linear minimum variance. Finally, based on the existing de-correlated filter, the algorithm of cross-covariance for de-correlated systems and the DF algorithm weighted by matrices, a de-correlated DF filtering algorithm is proposed. An example verifies the effectiveness of the proposed RLMVE-based DF filter.