Where did the children live more than half the nights of the year?
Only the parent where the children lived more than half the nights of the year can automatically claim the children as dependents. (This is the custodial parent, and the IRS only follows where the child actually physically lived, and does not follow court custody orders if the order is different from where the child lived.) The non-custodial parent can only claim a child if the custodial parent signs a waiver. Even if the custodial parent signs a waiver, the non-custodial parent can only claim the child tax credit. The custodial parent only can use the child to qualify for head of household, the dependent care credit, and EIC. Those benefits can never be waived, released or transferred to the non-custodial parent.
The custodial parent should list both children and indicate that they lived with the parent more than half the year (in turbotax, select "7 months" or longer). If not claiming one of the children, the custodial parents says "yes" to "I will sign a release and allow the other parent to claim this child." You must then sign the release form and give it to the other parent.
Then, the non-custodial parent will list the child and say the child lived with them less than half the year (choose "6 months" or less). The program will ask, "will you get a signed release form?" Answer yes. The non-custodial parent will claim the child tax credit but not EIC or dependent care benefits or head of household. After the non-custodial parent e-files, they must mail the signed release form given to them by the other parent to the IRS along with a cover page that Turbotax will tell them to print out.
Your situation is a symptom that someone did something wrong.
Either, you incorrectly claimed both children when you should have claimed one and signed a waiver for the other, or your ex is incorrectly trying to claim more benefits than allowed for a child who did not live with them, by claiming the child did live with them.
If your ex is the non-custodial parent, they need to delete the child and start over and answer the residency questions more carefully, and you need to give them the release form. Then they should be able to e-file. https://www.irs.gov/pub/irs-pdf/f8332.pdf
If your ex is the custodial parent and is allowing you to claim a child, then your ex should list both children, answer "more than half the year" to the residency questions, and give you a signed release form for the child you will claim. They will be blocked from e-filing (since you already filed) and will have to file by mail. Then you will need to file an amended tax return to remove the children and only claim the child that you have a release form for, and you will need to change your answers to the residency questions to indicate less than half the year.
