We present Champagne, a whole-genome method for generating character matrices for phylogenomic analysis using large genomic indel events. By rigorously picking orthologous genes and locating large insertion and deletion events, Champagne delivers a character matrix that considerably reduces homoplasy compared with morphological and nucleotide-based matrices, on both established phylogenies and difficult-to-resolve nodes in the mammalian tree. Champagne provides ample evidence in the form of genomic structural variation to support incomplete lineage sorting and possible introgression in Paenungulata and human-chimp-gorilla which were previously inferred primarily through matrices composed of aligned single-nucleotide characters. Champagne also offers further evidence for Myomorpha as sister to Sciuridae and Hystricomorpha in the rodent tree. Champagne harbors distinct theoretical advantages as an automated method that produces nearly homoplasy-free character matrices on the whole-genome scale.