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Impact assessment of natural hazards requires the consideration of both extreme and non-extreme events. Extensive research has been conducted on the joint modeling of bulk and tail in univariate settings; however, the corresponding body of research in the context of multivariate analysis is comparatively scant. This study extends the univariate joint modeling of bulk and tail to the multivariate framework. Specifically, it pertains to cases where multivariate observations exceed a high threshold in at least one component. We propose a multivariate mixture model that assumes a parametric model to capture the bulk of the distribution, which is in the max-domain of attraction (MDA) of a multivariate extreme value distribution (mGEVD). The tail is described by the multivariate generalized Pareto distribution, which is asymptotically justified to model multivariate threshold exceedances. We show that if all components exceed the threshold, our mixture model is in the MDA of an mGEVD. Bayesian inference based on multivariate random-walk Metropolis-Hastings and the automated factor slice sampler allows us to incorporate uncertainty from the threshold selection easily. Due to computational limitations, simulations and data applications are provided for dimension $d=2$, but a discussion is provided with views toward scalability based on pairwise likelihood., Comment: 34 pages, 7 figures