Rafael Benitez's side fielded the ineligible Denis Cheryshev against Cadiz, although he had accrued a one-game suspension while on loan at Villarreal.
The Royal Spanish Football Federation had already rejected an initial appeal against the decision, meaning at Real Madrid escalated it to TAD.
Real said they disagreed with the decision and would now be appealing through the courts.
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Real president Florentino Perez had previously said they did not break the rules and that neither Cheryshev nor the club were notified that he was banned.
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