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By extending single-species distribution models, multi-species distribution models and joint species distribution models are able to describe the relationship between environmental variables and a community of species. It is also possible to model either the marginal distribution of each species (multi-species models) in the community or their joint distribution (joint species models) under certain assumptions, but a model describing both entities simultaneously has not been available. We propose a novel model that allows description of both the joint distribution of multiple species and models for the marginal single-species distributions within the framework of multivariate transformation models. Model parameters can be estimated from abundance data by two approximate maximum-likelihood procedures. Using a model community of three fish-eating birds, we demonstrate that inter-specific food competition over the course of a year can be modeled using count transformation models equipped with three time-dependent Spearman's rank correlation parameters. We use the same data set to compare the performance of our model to that of a competitor model from the literature on species distribution modeling. Multi-species count transformation models provide an alternative to multi- and joint- species distribution models. In addition to marginal transformation models capturing single-species distributions, the interaction between species can be expressed by Spearman's rank correlations in an overarching model formulation that allows simultaneous inferences for all model parameters. A software implementation is available in the "cotram" add-on package to the R system for statistical computing.