The 32-year-old left-back had been without a club since being released by Premier League side West Ham at the end of last month, but has put pen to paper on a deal with the Madrid club until the end of the season.
Rat only joined West Ham in May after his contract with Shakhtar Donetsk expired and made 20 appearances for Sam Allardyce’s side before his stay at Upton Park was cut short.
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