A significant portion of genes in vertebrate genomes belongs to multigene families, with each family containing several gene copies whose presence/absence, as well as isoform structure, can be highly variable across individuals. Existing de novo techniques for assaying the sequences of such highly-similar gene families fall short of reconstructing end-to-end transcripts with nucleotide-level precision or assigning alternatively spliced transcripts to their respective gene copies. We present IsoCon, a high-precision method using long PacBio Iso-Seq reads to tackle this challenge. We apply IsoCon to nine Y chromosome ampliconic gene families and show that it outperforms existing methods on both experimental and simulated data. IsoCon has allowed us to detect an unprecedented number of novel isoforms and has opened the door for unraveling the structure of many multigene families and gaining a deeper understanding of genome evolution and human diseases.