  Ensemble Kalman--Bucy filters (EnKBFs) are an important tool in Data
Assimilation that aim to approximate the posterior distribution for continuous
time filtering problems using an ensemble of interacting particles. In this
work we extend a previously derived unifying framework for consistent
representations of the posterior distribution to correlated observation noise
and use these representations to derive an EnKBF suitable for this setting as a
constant gain approximation of these optimal filters. Existence and uniqueness
results for both the EnKBF and its mean field limit are provided. The existence
and uniqueness of solutions to its limiting McKean-Vlasov equation does not
seem to be covered by the existing literature. In the correlated noise case the
evolution of the ensemble depends also on the pseudoinverse of its empirical
covariance matrix, which has to be controlled for global well posedness. These
bounds may also be of independent interest. Finally the convergence to the mean
field limit is proven. The results can also be extended to other versions of
EnKBFs.
