Detecting structural variants (SVs) from sequencing data is a key problem in genome analysis, but the full diversity of SVs is not captured by most methods. We introduce the Automated Reconstruction of Complex Structural Variants (ARC-SV) method, which detects a broad class of structural variants from paired-end whole genome sequencing (WGS) data. Analysis of samples from NA12878 and HuRef suggests that complex SVs are often misclassified by traditional methods. We validated our results both experimentally and by comparison to whole genome assembly and PacBio data; ARC-SV compares favorably to existing algorithms in general and gives state-of-the-art results on complex SV detection. By expanding the range of detectable SVs compared to commonly-used algorithms, ARC-SV allows additional information to be extracted from existing WGS data.