Final answer:
The centromere is the part of sister chromatids that acts as the unit attaching them to microtubules during meiosis I, facilitated by the kinetochore protein complex.
Step-by-step explanation:
The part of the sister chromatids that act as a unit and attach at the same microtubule during meiosis I is the centromere. During the late prophase or prometaphase transition, microtubules from the mitotic spindle invade the former nuclear area where the nuclear membrane has broken down.
The kinetochore, a protein complex at the centromere, is the point of attachment between these microtubules and the sister chromatids. This ensures that when the microtubules pull the linked chromatids apart during anaphase I, the sister chromatids remain tightly bound together at the centromere while the homologous chromosomes are separated.