This is a standard situation, it only seems more complicated because the children are not living in the US.
First, your ex may not claim a child living outside the US as a dependent unless that child is a US citizen or permanent resident and has a Social Security number or green card, no matter if you follow the other rules described.
Then, these are the rules that apply to a child who is a US citizen or permanent resident. The only parent who is automatically entitled to claim the children as dependents is the parent where the children live more than half the nights of the year. That is the custodial parent in IRS terminology. The IRS does not follow court orders or “50-50 custody“, only where the children live more than half the nights of the year.
The custodial parent may transfer the dependent claim to the noncustodial parent by giving the noncustodial parent a signed form 8332, that specifies which years the non-custodial parent may claim the child or children. This form allows the noncustodial parent to claim the $2000 or $500 dependent tax credit, but it does not allow the noncustodial parent to use the children to qualify for head of household status, earned income credit, or the child care credit. These tax benefits always stay with the custodial parent and can never be waived, transferred, or shared. This is true even if the custodial parent (you) is not a US taxpayer.
What you are supposed to do is to give the other parent a signed form 8332 allowing them to claim the children as dependents (only the children who have Social Security numbers or green cards). The other parent is then supposed to indicate on their tax return that the children do not live with them more than half the nights of the year and they are only claiming a dependent because of form 8332. The other parent will be required to mail the IRS the original signed form 8332, either attached to their tax return, or separately with a cover letter after e-filing.
If the other parent has been claiming the children as if they lived with him all year in the US, without form 8332, that is incorrect, and he may be claiming excess tax benefits that he is not entitled to. However, because you do not file a US tax return, there is no automatic mechanism for this problem to be discovered. You could file a fraud report with the IRS if you wanted to. To make sure that you do everything that is legally correct, you should be sending your ex a signed copy of form 8332 for each year that your ex is allowed to claim one or more of your children as dependents. If your ex breaks the law After that, it’s not your problem, unless you want to get your ex in trouble.
