Chiasmata form during the prophase I stage of meiosis, specifically during the pachytene phase. The process begins with synapsis, where homologous chromosomes align closely along their length. The exchange of genetic material is facilitated by the breaking and rejoining of DNA strands at corresponding positions on homologous chromosomes. This exchange results in physical connections known as chiasmata, which are crucial for the stability of the paired chromosomes until anaphase I.