We introduce Scallop, an accurate, reference-based transcript assembler for RNA-seq data. Scallop significantly improves reconstruction of multi-exon and lowly expressed transcripts. On 10 human samples aligned with STAR, Scallop produces (on average) 35.7% and 37.5% more correct multi-exon transcripts than two leading transcript assemblers, StringTie and TransComb, respectively. For transcripts expressed at low levels in the same samples, Scallop assembles 65.2% and 50.2% more correct multi-exon transcripts than StringTie and TransComb, respectively. Scallop obtains this improvement through a novel algorithm that we prove preserves all phasing paths from reads (including paired-end reads), while also producing a parsimonious set of transcripts and minimizing coverage deviation.