What holds non-sister chromatids of homologous chromosomes together in prophase i of meiosis?

The correct answer is: The synaptonemal complex.

Non-sister chromatids are chromatids from two homologous chromosomes. In prophase I of meiosis two homologous chromosomes pair and this pairing is called synapsis. This allows possible crossing over to occur. Two non-sister chromatids are connected in synapsis by a protein-RNA complex called the synaptonemal complex.

Crossing-over or the physical exchange of chromosome parts might occur and bring to genetic recombination. The sites of crossing-over between nonsister chromatids and are called chiasmata.