The start that Real made seemed like they had turned the corner, taking the game to City and going in front courtesy of Rodrygo’s low drilled effort.
But they conceded twice in the final 10 minutes of the first half through Nico O’Reilly poking in from close range and Erling Haaland’s penalty.
The Real ultras dressed all in white behind the goal had been chanting and clapping until that point, but when the Norwegian’s spot-kick rippled the net in front of them they were silenced.
Ex-Bayer Leverkusen boss Alonso took over before the Club World Cup in the summer and started this season by winning 13 of his first 14 games, but it started to go wrong after the loss at Liverpool on 4 November.
Real have won just twice in eight games since and there has been an apparent clash of cultures at the club with players seemingly not buying into his strict methods and wanting to adopt a pressing style of play.
Asked if his players had played for him, Alonso said: “The performance has been intense. A professional Champions League level game. They have given their best, so from my side no complaints for anything.
“It is hard to take another defeat but we tried until the end.”
England international Jude Bellingham told TNT Sport that the squad were “100%” behind their manager.
“I personally have a great relationship with him, a lot of the lads do as well,” he added. “After the first run of games we had some really great conversations internally and we felt we were on the back side of that form.
“In the past couple of games we let ourselves down again. No-one’s downing tools, no-one’s complaining and moaning thinking the season is over.”