You can both file as head of household if you both meet the requirements for that filing status. One of the requirements is that "a qualifying person lived with you in the home for more than half the year." That requirement is based on where the child actually lived. Your divorce agreement, or any informal agreement that you make, has nothing to do with it.
There is no such thing as 50/50 custody in tax law. Whichever parent the child lived with for more nights during the year is the custodial parent. Among the benefits of being the custodial parent is the ability to file as head of household. Since you have two children, you and your ex could arrange the children's time in your homes so that each of you is the custodial parent of one of the children. That way you could both file as head of household. That would also make it easy to satisfy the requirements of your divorce agreement to each claim one child as a dependent each year. You would each claim the child for whom you are the custodial parent. You would be advised to keep careful records of where each child slept each night. That would eliminate any disputes, and also help to convince the IRS of where each child lived, if necessary.
At some point someone is going to tell you about IRS Form 8332. That form is used to let the custodial parent allow the noncustodial parent to claim the child as a dependent. But it does not allow the noncustodial parent to file as head of household. Head of household is always based on "physical location," as you put it. If you arrange things so that each of you is the custodial parent of one child, as I suggested, you have no need for Form 8332.
