Faithful chromosome segregation requires that the sister chromatids be disjoined completely. Defective disjunction can lead to the persistence of histone-free threads of DNA known as ultra-fine bridges (UFBs) that connect the separating sister DNA molecules during anaphase. UFBs arise at specific genomic loci and can only be visualized by detection of associated proteins such as PICH, BLM, topoisomerase IIIα, and RPA. However, it remains unknown how these proteins work together to promote UFB processing. We used a combination of ensemble biochemistry and new single-molecule assays to reconstitute key steps of UFB recognition and processing by these human proteins in vitro. We discovered characteristic patterns of hierarchical recruitment and coordinated biochemical activities that were specific for DNA structures modeling UFBs arising at either centromeres or common fragile sites. Our results describe a mechanistic model for how unresolved DNA replication structures are processed by DNA-structure-specific binding factors in mitosis to prevent pathological chromosome nondisjunction.