Understanding the mechanisms that maintain microbial biodiversity is a critical aspiration in ecology. Past work on microbial coexistence has largely focused on species pairs, but it is unclear whether pairwise coexistence in isolation is required for coexistence in a multispecies community. To address this question, we conducted hundreds of pairwise competition experiments among the stably coexisting members of 12 different enrichment communities in vitro. To determine the outcomes of these experiments, we developed an automated image analysis pipeline to quantify species abundances. We found that competitive exclusion was the most common outcome, and it was strongly hierarchical and transitive. Because many species that coexist within a stable multispecies community fail to coexist in pairwise co-culture under identical conditions, we concluded that multispecies coexistence is an emergent phenomenon. This work highlights the importance of community context for understanding the origins of coexistence in complex ecosystems.