HOH is strictly based on where the child lives more than half the nights of the year. In order for both parents to claim HOH status, you have to arrange actual physical custody so that child A lives with you at least 183 nights, and child B lives with your ex at least 183 nights.
In normal years with 365 days, more than half means 183 or more nights and less than half means 182 or fewer nights. It only has to be 1 day different (or more), but HOH is strictly based on physical presence more than half the year. In leap years with 366 days, more than half means 184 or more nights. If each parent has custody exactly 183 nights of a leap year, then no parent can claim HOH because no parent had custody more than half (because exactly half is not more than half).
Note that if you arrange actual physical custody of the children so that each parent has "more than half" physical custody of a child, then a form 8332 dependent release is not required of either parent. If one parent has physical custody of both children for 183 or more nights, the other parent can't claim HOH, and can't claim the child as a dependent without a form 8332.
For 2021, you can claim HOH based on having custody more than half the nights of the year. Your ex can only claim the children as dependents if you give your ex a signed form 8332 dependent release. In that case, you will tell the tax program that you had custody but you are signing the form; you will receive HOH but not the child tax credit. Your ex will say they had custody less than half the year but did get a form 8332; their program will give them the child tax credit but not HOH. If your ex tries to claim the children without form 8332 by saying they lived with him more than half the year, you may not be able to e-file and both of you will get letters from the IRS investigating the situation.
