Protein–protein interactions (PPIs) play essential roles in most biological processes. The binding interfaces between interacting proteins impose evolutionary constraints that have successfully been employed to predict PPIs from multiple sequence alignments (MSAs). To construct MSAs, critical choices have to be made: how to ensure the reliable identification of orthologs, and how to optimally balance the need for large alignments versus sufficient alignment quality. Here, we propose a divide-and-conquer strategy for MSA generation: instead of building a single, large alignment for each protein, multiple distinct alignments are constructed under distinct clades in the tree of life. Coevolutionary signals are searched separately within these clades, and are only subsequently integrated using machine learning techniques. We find that this strategy markedly improves overall prediction performance, concomitant with better alignment quality. Using the popular DCA algorithm to systematically search pairs of such alignments, a genome-wide all-against-all interaction scan in a bacterial genome is demonstrated. Given the recent successes of AlphaFold in predicting direct PPIs at atomic detail, a discover-and-refine approach is proposed: our method could provide a fast and accurate strategy for pre-screening the entire genome, submitting to AlphaFold only promising interaction candidates—thus reducing false positives as well as computation time.