Recent works in the literature have proposed models and algorithms for nonparametric estimation of finite multivariate mixtures. In these works, independent coordinates conditional on the subpopulation from which each observation is drawn is assumed, so that the dependence structure comes only from the mixture. Here this assumption is relaxed, allowing for independent multivariate blocks of coordinates, conditional on the subpopulation from which each observation is drawn. Otherwise the blocks 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 evaluated 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.