You could each list the property on your return as 50% owner and claim half of everything. I think you are thinking you took the first half and he can have the second half. That is not how this works.
Do a partnership return with the rental house and issue a k-1 to each of you. When you do the partnership return, you enter how much of the house each person owns. Pick up the depreciation right where you are so the IRS can follow the trail, if needed. See IRS Tax information for Partnerships. This makes the most sense to me.
Edited 1/21/2022 (8:09 PST)
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