Absent an agreement of the parties, Federal tax code Publication 504 suggests that the parent exercising more than 50% of the time-sharing would receive the tax benefit.
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For there to be a modification of the Final Judgment, you would need to show that there had been a substantial, unanticipated change in circumstances. I am not sure that this situation would rise to that level. Have you asked your former spouse if they would be willing to allow you to claim one of the children each year?
Typically the only things you can modify post final judgment are a) amount of child support, b) amount of spousal support, and c) parenting plan. His request does not fall into any category, unfortunately, you may have given this away.
I would speak with the attorney who did your divorce and ask him why it was not addressed.
In order to modify a decree you need to prove that there was a substantial, material and unanticipated change in circumstances. If the decree is silent regarding claiming the children on tax returns, then typically the "custodial" or majority time sharing parent may claim.
I don't see how you could claim this was unanticipated if it was not addressed at all. However, you might try one thing. Because of the tax deduction and tax credits go to the "custodial parent" it might be worthwhile to recalculate child support based upon that fact. With the deductions and credits the "custodial parent" gets, her net income may be higher than what you thought when the case resolved. That would permit you to recalculate your child support.
This tactic would not get you to be able to claim the children on your taxes but it might give you some relief on the support.
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