The job is not shared fairly if one person must keep asking whether it has been done.

Following up is work. It never appears in any division of labor. When one person holds the mental list of what is outstanding, checks on it and raises it again, they are doing a second job on top of their share, even when the tasks themselves are split evenly. The other person experiences this as nagging rather than as evidence of imbalance. A task is only really shared when it lives in someone else’s head, not only in their hands.

Pick one recurring thing and take it over completely, including remembering it. If you are the one chasing, hand the whole item across rather than only the doing.