The European champions were beaten 1-0 by Espanyol on Saturday.

They remain top of La Liga having won 15 of their opening 22 fixtures.

Atletico Madrid are only one point behind their city rivals, while FC Barcelona are four points off first.

Carlo Ancelotti's side had a ridiculous 77 per cent possession and 21 shots to Espanyol's five, with the hosts committing 11 more fouls but only receiving two bookings.

The match-winner was controversially Carlos Romero, who was earlier booked for a challenge on Kylian Mbappe that many people felt should have been a straight red card for bringing down the Frenchman from behind on the counter.

Real Madrid manager Ancelotti slammed the refereeing after the defeat, saying: "The decision the referee and the VAR took is inexplicable. Everyone saw it.

"The most important thing is to protect the player. It's a clear foul, a very ugly challenge.

"Fortunately nothing happened in terms of an injury, but the VAR is there for that. It's inexplicable to us that he didn't show a red card."

Speaking after the match, defender Romero revealed that he apologised to Mbappe.

"I knew that it's impossible to stop [Mbappe] on the run," he said.

"I did what I could to stop him. The challenge was a bit ugly, I didn't like that, and I said sorry to him."

Madrid are still absolutely fuming days later and have let their feelings about the refereeing known to the Spanish Football Federation (RFEF) and Spain's Sports Ministry (CSD), lodging a formal complaint.

As per a report from ESPN, Los Blancos believe the game "represents the culmination of a completely discredited refereeing system" with "a level of manipulation and adulteration which cannot be ignored."

Madrid wrote a four-page open letter on Monday, revealing the club's hierarchy would make a formal complaint over referee Alejandro Muniz Ruiz's and Video Assistant Referee Javier Iglesias Villanueva.

The club complained about the "refereeing scandal" and demanded a "structural reform".

"The events that occurred in this match exceeded any margin for human error or refereeing interpretation," the letter said.

"What happened at the RCDE Stadium represents the culmination of a completely discredited refereeing system, in which decisions against Real Madrid have reached a level of manipulation and adulteration of the competition that can no longer be ignored."

On the foul on Mbappe, the letter continued: "It is particularly serious that in his report, the referee recorded that the action occurred 'while contesting the ball'.

"Thus distorting and falsifying the reality of what happened, with the only imaginable purpose of justifying his arbitrary decision… The referee's decision, and the inaction of the VAR, are of unquestionable seriousness.

"What is really needed is structural reform which prevents these events from happening again and again, as has been happening repeatedly to the detriment of Real Madrid and the integrity of the league.

"This refereeing scandal is not an isolated case. Superficial changes are not enough. The refereeing system must be completely renewed, from its structure to those within it."

The RFEF responded with a statement of their own, defending the refereeing integrity in La Liga.

They said: "The task of refereeing, by its very nature, is subject to review and analysis, but this cannot lead to generalized accusations that call into question their integrity, since this not only affects the referees themselves, but also erodes the credibility of football as a whole.

"It is important to reflect on the consequences of this type of systematic questioning of refereeing. Constantly delegitimizing the work of referees outside of established channels generates a climate of mistrust that benefits neither Spanish football nor its competitions.

"Constructive criticism is always welcome and necessary for continuous improvement, but it must be carried out within the established frameworks and with the respect required by all parties that make up our sport."

Real Madrid make formal complaint over La Liga refereeing in Espanyol loss