The Reds manager admitted his team deserved to lose the first leg of the Champions League quarterfinal encounter but was furious Lucas Vazquez's first-half challenge on Mane that preceded Real Madrid's second goal went unpunished.
He made his feelings clear to Brych at halftime and after the final whistle.
"The situation with Sadio - what the ref did tonight I don't understand," Klopp said, as reported by The Guardian's Andy Hunter. "For me, that was something personal because he dealt with the situation with Sadio, which was a clear foul, like he was a diver or whatever.
"From that moment on, whenever Sadio went down, he didn't get anything. That is not right. That is what I told (Brych) after the game, that I thought he was unfair with Sadio."
Mane drew one foul in the match at the Estadio Alfredo Di Stefano. He's been fouled an average of 2.5 times per 90 minutes in the Premier League this season.
Liverpool will try to fight back from the 3-1 aggregate deficit when Real Madrid travel to Anfield on April 14.