Recently developed methods that utilize partitioning of long genomic DNA fragments, and barcoding of shorter fragments derived from them, have succeeded in retaining long-range information in short sequencing reads. These so-called read cloud approaches represent a powerful, accurate, and cost-effective alternative to single-molecule long-read sequencing. We developed software, GROC-SVs, that takes advantage of read clouds for structural variant detection and assembly. We apply the method to two 10x Genomics data sets, one chromothriptic sarcoma with several spatially separated samples, and one breast cancer cell line, all Illumina-sequenced to high coverage. Comparison to short-fragment data from the same samples, and validation by mate-pair data from a subset of the sarcoma samples, demonstrate substantial improvement in specificity of breakpoint detection compared to short-fragment sequencing, at comparable sensitivity, and vice versa. The embedded long-range information also facilitates sequence assembly of a large fraction of the breakpoints; importantly, consecutive breakpoints that are closer than the average length of the input DNA molecules can be assembled together and their order and arrangement reconstructed, with some events exhibiting remarkable complexity. These features facilitated an analysis of the structural evolution of the sarcoma. In the chromothripsis, rearrangements occurred before copy number amplifications, and using the phylogenetic tree built from point mutation data we show that single nucleotide variants and structural variants are not correlated. We predict significant future advances in structural variant science using 10x data analyzed with GROC-SVs and other read cloud-specific methods.