Without several first-team stars such as Karim Benzema, Sergio Ramos, Daniel Carvajal and Eden Hazard, Los Blancos struggled to break down La Real.
The draw leaves Real Madrid in third place on the La Liga standings, five points behind Atletico Madrid, who host them next week.
Here are the Real Madrid player ratings from their draw against Real Sociedad.
Real Madrid player ratings against Real Sociedad
Thibaut Courtois - 7/10
There was no way Thibaut Courtois could have kept out that looping header from Portu. Apart from that goal, the Belgian was sound between the sticks and made two saves in the match.
Lucas Vazquez - 7/10
Arguably the best player for Real Madrid on Monday, Lukas Vazquez created chance after chance with excellent crosses inside the area. He also took up good attacking positions in Real Sociedad's half.
Raphael Varane - 7/10
With three recoveries and four clearances, Raphael Varane was vital in the defense for Real Madrid. He used his positional awareness and strength to nullify in-form striker Aleksander Isak.
Nacho - 7/10
Real Madrid's utility man covered well for Ferland Mendy when he was caught upfield and dealt with Portu with aplomb. The Spaniard registered three tackles and three clearances during the game.
Ferland Mendy - 6/10
Ferland Mendy was a huge threat from the left wing but his crosses left a lot to be desired. He was also beaten in the air by Portu for the opening goal.
Luka Modric - 7/10
Luka Modric got into some pockets of space and mustered a few decent shots at goal. However, he couldn't really influence the game and was taken off in the second half.
Casemiro - 6/10
While he made decent defensive contributions, Casemiro had a hard time at the other end of the field, making several wayward attempts at goal. This resulted in him wasting a few good chances to score.
Toni Kroos - 7/10
The veteran German midfielder was key to Real Madrid's ball progression, constantly spraying passes all over the field. He showed excellent commitment in trying to regain the ball when not in possession.
Marco Asensio - 5/10
Marco Asensio seems to have no future at Real Madrid. With the likes of Karim Benzema, Eden Hazard, Rodrygo and Federico Valverde all injured, Asensio hasn't taken his chances to impress. He was below his best once again against Real Sociedad, even missing a great chance towards the end.
Mariano - 5/10
This was another disappointing performance from the Dominican striker. He led the line for Real Madrid in the absence of Benzema but failed to truly test La Real.
Isco - 7/10
Isco was lively in the attack, creating several chances and linking up brilliantly with Toni Kroos and Ferland Mendy. Zinedine Zidane will hope that he can keep up this form till the end of the season.
Vinicius Junior - 7/10
Vinicius Junior came off the bench for his 100th appearance in a Real Madrid shirt and marked the milestone with an important goal.
Rodrygo - 6/10
Rodrygo brought energy to the game after coming on and made one key pass.
Hugo Duro - 6/10
The 21-year old played for over 30 minutes but his impact was minimal.
Federico Valverde - 7/10
Federico Valverde played a key role in the build-up to Real Madrid's goal. He would be content with his contribution.