For tax purposes, there is no such thing as joint custody (50/50), regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody.
Yes, the IRS expects you to count the numbers of nights the child sleeps at each parent's home. It is allowed for you to arrange the children's schedules so that one child spends more than half the year with the parent that is claiming him every other year.
In the rare case (could probably only happen in a leap year like 2020), where the time that each parent has the child is exactly equal, then the parent with the higher income (AGI) is the custodial parent, for the purpose of determining who has first priority on claiming the child as a dependent. But then neither parent can claim a Qualifying Child dependent, for some tax benefits because neither parent had the child the required MORE than half the year. (no earned income credit, based on that child, and the child would not qualify the parent for Head of Household filing status).
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.
So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
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