Abstract Many processes of biological diversification can simultaneously affect multiple evolutionary lineages. Examples include multiple members of a gene family diverging when a region of a chromosome is duplicated, multiple viral strains diverging at a “super-spreading” event, and a geological event fragmenting whole communities of species. It is difficult to test for patterns of shared divergences predicted by such processes, because all phylogenetic methods assume that lineages diverge independently. We introduce a Bayesian phylogenetic approach to relax the assumption of independent, bifurcating divergences by expanding the space of topologies to include trees with shared and multifurcating divergences. This allows us to jointly infer phylogenetic relationships, divergence times, and patterns of divergences predicted by processes of diversification that affect multiple evolutionary lineages simultaneously or lead to more than two descendant lineages. Using simulations, we find the new method accurately infers shared and multifurcating divergence events when they occur, and performs as well as current phylogenetic methods when divergences are independent and bifurcating. We apply our new approach to genomic data from two genera of geckos from across the Philippines to test if past changes to the islands’ landscape caused bursts of speciation. Unlike our previous analyses restricted to only pairs of gecko populations, we find evidence for patterns of shared divergences. By generalizing the space of phylogenetic trees in a way that is independent from the likelihood model, our approach opens many avenues for future research into processes of diversification across the life sciences. Significance statement Phylogenetic models have long assumed that lineages diverge independently. Processes of diversification that are of interest in biogeography, epidemiology, and genome evolution, violate this assumption by affecting multiple evolutionary lineages. To relax the assumption of independent divergences and infer patterns of divergences predicted by such processes, we introduce a new way of conceptualizing, modeling, and inferring phylogenetic trees. We apply the new approach to genomic data from geckos distributed across the Philippines, and find support for patterns of shared divergences predicted by repeated fragmentation of the archipelago by interglacial rises in sea level.