The rules for determining filing status are not the same as the rules determining which parent can claim a child dependent to get the various tax benefits associated with who gets to claim which child as a dependent. Under the special rules for children of divorced or separated parents, only the parent where the child lives more than half the nights of the year can use the child as a qualifying person for head of household (HOH) status. There is no release mechanism for splitting HOH filing status. Only one parent can claim it based on the physical presence test mentioned above. Even in a 50-50 custody split, one parent only needs to have the child for one night more than the other parent to satisfy the test for using HOH status.
So for the IRS, there is no 50% custody, you have to count nights, and one parent will almost always have more than the other. If for some reason the # of days is exactly equal (like in a leap year, 366 days, exactly 183 nights for each parent) then the tax benefits are assigned to the parent with the higher income.
If the divorce decree says that the non-custodial parent (parent with less than 50% time) gets the dependents in a certain year, the custodial parent must fill out and sign a copy of form 8332 and give it to the other parent, this releases the tax benefits to the other parent. However, the form 8332 only transfers the dependent exemption and the child tax credit. Eligibility for Head of Household, Dependent Care Credit, and Earned Income Credit (EIC) always stays with the custodial parent and can not be transferred.
There are a few rare cases where it could be possible for both parents to file as Head of Household in the same tax year. For example, if there are two children, one child spends 51% of the year with one parent, and the second child spends 51% of the year with the other parent, both parents may be able to file as Head of Household in the same tax year. If you feel like you and your ex-spouse qualify for this exception, you will need to keep very careful records of where the children spend their days and nights throughout the year.
You should keep a “child custody log” because there is a good chance that both parents filing as HOH post-divorce will trigger an audit by the IRS. But there is nothing guaranteeing that a child custody log by itself will satisfy the IRS in the event of an audit. The IRS could request additional information to determine that the 51% time requirement was met by each parent.