Learning the structure of probabilistic graphical models for complex real-valued domains is a formidable computational challenge. This inevitably leads to significant modelling compromises such as discretization or the use of a simplistic Gaussian representation. In this work we address the challenge of efficiently learning truly expressive copula-based networks that facilitate a mix of varied copula families within the same model. Our approach is based on a simple but powerful bivariate building block that is used to highly efficiently perform local model selection, thus bypassing much of computational burden involved in structure learning. We show how this building block can be used to learn general networks and demonstrate its effectiveness on varied and sizeable real-life domains. Importantly, favorable identification and generalization performance come with dramatic runtime improvements. Indeed, the benefits are such that they allow us to tackle domains that are prohibitive when using a standard learning approaches.