My ex claimed my children despite them spending the majority of nights in my home. How do I fight this?

If they spent more nights with you, you have the right to claim them. You can claim them and mail your return since it will be rejected if she files first. The IRS will send both of you letters to sort it out. The loser will pay tax, penalty and interest. 

The custody agreement doesn't count, what counts is where the children actually physically spent the night, and how many nights in each home.

If you are the parent who had custody 183 or more nights, then you are the only parent automatically entitled to claim the children as dependents.

If this is for 2023, your ex hasn't actually filed yet, since the IRS is not accepting tax returns until January 29.  She might have submitted a return, but it hasn't been processed yet.  You can still file and try and claim both children, and see whose return gets processed first.  If your return is rejected, print the return and mail it in with the correct dependent claim.  The IRS will process the duplicate dependent claim and will eventually send letters to both parents to investigate.  You may need to be able to prove where the children actually lived, not just where the court said they should live.   You may need a diary, calendar or text messages showing the exchanges, and so on, that establish where the children actually spent the nights. 

If this is for 2023, and you already filed claiming one child, you must wait and see what happens when the IRS opens.  If your return is rejected, add the second child and print and mail your return.  If your return is accepted, you will have to wait another 6 weeks to file an amended return to claim the second child.

If this was for 2022 (last year's tax filing season), then you can file an amended return to claim the child.

In all cases, expect the IRS to investigate the claims of both parents.