Faithful DNA replication is vital to prevent disease-causing mutations, chromosomal aberrations and malignant transformation. However, accuracy conflicts with pace and flexibility and cells rely on specialized polymerases and helicases to ensure effective and timely replication of genomes that contain DNA lesions or secondary structures. If and how cells can tolerate a permanent barrier to replication is, however, unknown. Here we show that a single unresolved G-quadruplexed DNA structure can persist through multiple mitotic divisions without changing conformation. Failed replication across a G-quadruplex causes single-strand DNA gaps that give rise to DNA double-strand breaks in subsequent cell divisions, which are processed by polymerase theta (POLQ)-mediated alternative end joining. Lineage tracing experiments further reveal that persistent G-quadruplexes cause genetic heterogeneity during organ development. Our data demonstrate that a single lesion can cause multiple unique genomic rearrangements, and that alternative end joining enables cells to proliferate in the presence of mitotically inherited replication blocks.