We develop a new formulation of nonanticipative rate distortion function (NRDF) to characterize and compute multidimensional partially observable Gauss-Markov processes with MSE distortion. The key result to obtain this new formulation is a "genie-aided" design of our decoder that encapsulates both its previous decoding symbols and the past observation symbols. The new formulation is applied to a system modeled by jointly Gaussian processes to obtain the following new results. (i) An optimal characterization of a new finite dimensional optimization problem and its corresponding optimal realization. Surprisingly, the information structure of the optimal realization reveals that the decoder is in fact independent of all the previous observations symbols. (ii) For time-invariant processes, we convexify our characterization under the assumption that all matrices commute by pairs and derive strong structural properties for the involved matrices for which our assumption is valid. (iii) We solve the convex program using KKT conditions to obtain a solution via a general reverse-waterfilling algorithm which demonstrates that the distortion allocation at each dimension can be computed by a third-degree polynomial equation.