Final answer:
Homologous chromosomes pair up during Prophase I of meiosis I, wherein recombination or crossing-over occurs, allowing alleles to reshuffle and increase genetic diversity.
Step-by-step explanation:
The pairs of homologous chromosomes come together during Prophase I of meiosis I. During this stage, the chromosomes first replicate and then synapse, aligning their like genes with each other. This alignment allows for recombination, or crossover, where segments of genetic material are exchanged between the homologous chromosomes. Through recombination, alternative forms of genes, or alleles, may result in the combination of maternal and paternal alleles onto the same chromosome, causing a mixing of genetic information. The process is instrumental for genetic diversity in sexually reproducing organisms, as each gamete produced will have a unique set of genetic information.