As shown during the SARS-CoV-2 pandemic, phylogenetic and phylodynamic methods are essential tools to study the spread and evolution of pathogens. One of the central assumptions of these methods is that the shared history of pathogens isolated from different hosts can be described by a branching phylogenetic tree. Recombination breaks this assumption. This makes it problematic to apply phylogenetic methods to study recombining pathogens, including, for example, coronaviruses. Here, we introduce a Markov chain Monte Carlo approach that allows inference of recombination networks from genetic sequence data under a template switching model of recombination. Using this method, we first show that recombination is extremely common in the evolutionary history of SARS-like coronaviruses. We then show how recombination rates across the genome of the human seasonal coronaviruses 229E, OC43 and NL63 vary with rates of adaptation. This suggests that recombination could be beneficial to fitness of human seasonal coronaviruses. Additionally, this work sets the stage for Bayesian phylogenetic tracking of the spread and evolution of SARS-CoV-2 in the future, even as recombinant viruses become prevalent.