Recent works in the literature have proposed models and algorithms for nonparametric estimation of finite multivariate mixtures. In these works, the model assumes independent coordinates, conditional on the subpopulation from which each observation is drawn, so that the dependence structure comes only from the mixture. Here, we relax this assumption, allowing in the multivariate observations independent multivariate blocks of coordinates conditional upon knowing which mixture component from which they come. Otherwise their density functions are completely multivariate and nonparametric. We propose an EM-like algorithm for this model, and derive some strategies for selecting the bandwidth matrix involved in the nonparametric estimation step of it. The performance of this algorithm is through several numerical simulations. We also experiment this new model and algorithm on an actual dataset from the model based, unsupervised clustering perspective, to illustrate its potential.