Los Blancos missed the chance to close the title gap on league leaders Atletico Madrid, with a five-point gap behind Diego Simone, having played one more game.
Cristian Portu put Real Sociedad ten minutes after the restart of the second stanza with former Arsenal Nacho Monreal’s excellent cross which beat Ferland Mendy to head home.
Zidane’s side did finally snatch something with two minutes to go as substitute Vinicius Junior deflected strike flew past Alex Remiro.
The 1998 FIFA World Cup winner told reporters: "We have to stay calm. We played well and created three or four clear chances but were up against a side that also created chances. Now we need to be patient, keep our cool and rest.
"We deserved much more when you look at all the opportunities we created. We dominated the game, but then there have been games where we won but didn’t deserve to. That’s football.”
"It was those 10-15 minutes and we changed it again and we were better playing a 4-3-3.Our substitutes did well after coming on.
"If you've asked me and that's now three times, then yes, maybe," Zidane replied when asked whether his changes had harmed Real Madrid.
"We had to change things because, after an hour, the team was a bit tired. Sometimes we have to change things.
"In the end, you try to find things to change the dynamic."
The draw ended a five-match winning streak in all competitions for Real Madrid ahead of derby trip to Atletico this weekend, with another defeat potentially spelling the end of their title defence.
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