This is a tricky situation that could cause damage in the family so think about what you are doing before you do anything. You may need to have a serious talk with the mother and educate her on the consequences of her actions.
Now do you have the right to claim the kids ... yes.
Could you paper file the return to assert the claim ... yes.
Will you get the refund for the kids ... yes.
However, the IRS will send out notices to both parties in a couple of years asking the person in the wrong to amend their return to un-claim the kids. If neither owns up to the wrong doing then the IRS will send audit letters to both parties and then you will need to prove your right to claim the kids so prepare now ... get out a calendar and start marking down which days the kids were with you and be prepared to defend your claim in 3 to 4 years from now. The loser will be required to pay back all the money the kids got for them + penalties + interest + the restriction to not claim the EIC for 10 years.
