MOTIVATION: Structural variants (SVs) play an important role in evolutionary and functional genomics but are challenging to characterize. High-accuracy, long-read sequencing can substantially improve SV characterization when coupled with effective calling methods. While state-of-the-art long-read SV callers are highly accurate, further improvements are achievable by systematically modeling local haplotypes during SV discovery and genotyping. RESULTS: We describe sawfish, an SV caller for mapped high-quality long reads incorporating systematic SV haplotype modeling to improve accuracy and resolution. Assessment against the draft Genome in a Bottle (GIAB) SV benchmark from the T2T-HG002-Q100 diploid assembly shows that sawfish has the highest accuracy among state-of-the-art long-read SV callers across every tested SV size group. Additionally, sawfish maintains the highest accuracy at every tested depth level from 10- to 32-fold coverage, such that other callers required at least 30-fold coverage to match sawfish accuracy at 15-fold coverage. Sawfish also shows the highest accuracy in the GIAB challenging medically relevant genes benchmark, demonstrating improvements in both comprehensive and medically relevant contexts.When joint-genotyping seven samples from CEPH-1463, sawfish has over 9000 more pedigree-concordant calls than other state-of-the-art SV callers, with the highest proportion of concordant SVs (81%). Sawfish's quality model enables selection for an even higher proportion of concordant SVs (88%), while still calling nearly 5000 more pedigree-concordant SVs than other callers. These results demonstrate that sawfish improves on the state-of-the-art for long-read SV calling accuracy across both individual and joint-sample analyses. AVAILABILITY AND IMPLEMENTATION: Sawfish source code, pre-compiled Linux binaries, and documentation are released on GitHub: https://github.com/PacificBiosciences/sawfish.