Zinedine Zidane's side created few chances on a wet night in the Spanish capital as stubborn Betis stood firm to claim a fifth successive draw.
Leaders Atletico remain two points clear of Real with a game in hand ahead of Sunday's visit to Athletic Bilbao.
Betis have been troublesome opponents to Real in the past with two wins and a draw on their last three visits.
Real struggled to fashion chances in a first half which saw both defences in full control on a slick surface.
Isco wasted a promising free-kick position with a wild attempt and Real's only on-target attempt in the first half came after 24 minutes.
The contest was more open after the interval with Marco Asensio trying his luck from distance.
Rodrygo's mishit cross then struck the crossbar, with Bravo appearing to get the faintest of touches on the ball.
It would have been a bizarre way for Rodrygo to register his first goal of the season, but Betis began to grow in confidence and carry a genuine threat at the other end.
Guido Rodriguez was left unmarked on the edge of the area to fire straight at Thibaut Courtois before Iglesias passed up another clearcut opportunity on the penalty spot.
But Betis were rarely troubled in the closing stages as Real suffered a damaging blow to the defence of their league title.