Motivated by the challenges in analyzing gut microbiome and metagenomic data, this paper introduces a novel mixture distribution for multivariate counts and a regression model built upon it. The flexibility and interpretability of the proposed distribution accommodate both negative and positive dependence among taxa and are accompanied by numerous theoretical properties, including explicit expressions for inter- and intraclass correlations, thereby providing a powerful tool for understanding complex microbiome interactions. Furthermore, the regression model based on this distribution facilitates the clear identification and interpretation of relationships between taxa and covariates by modeling the marginal mean of the multivariate response (i.e., taxa counts). Inference is performed using a tailored Hamiltonian Monte Carlo estimation method combined with a spike-and-slab variable selection procedure. Extensive simulation studies and an application to a human gut microbiome dataset highlight the proposed model's substantial improvements over competing models in terms of fit, interpretability, and predictive performance.