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Xabi Alonso before a match
Alonso has signed a three-year contract until June 2028 and the 43-year-old replaces Carlo Ancelotti, who took charge of his last game on Saturday.
Alonso had been in charge of Bayer Leverkusen and confirmed he would be leaving the German club this month.
The Spaniard took charge of Leverkusen in October 2022 and guided them to the double last season as they lifted their first Bundesliga title, without losing a game. They also won the German Cup and reached the Europa League final.
He had a contract at Leverkusen until 2026 but the club said they had granted his wish to terminate his deal at the end of the season.
Alonso will be presented as Real coach on Monday and will take charge of Real ahead of the Club World Cup which starts in the United States next month.