This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply. Olga Carmona opened the scoring after just 10 minutes for the hosts but Chelsea equalised before half-time through Niamh Charles, before Sam Kerr put the visitors ahead with a powerful header with 15 minutes left on the clock.
But minutes later Real Madrid were given a route back into the match in controversial fashion, with referee Frida Klarlund awarding a penalty for a foul on Athenea del Castillo even though the tackle happened outside the box.
Despite the pleas of the Chelsea players, and with no VAR in use to intervene, Carmona stepped up to bury the spot-kick and level the scores.
Chelsea rallied despite that setback, with Lauren James striking the crossbar, and then appeared to have bagged a last-gasp winner when Charles volleyed home a deep cross - only to see the offside flag raised.
But replays of the incident quickly revealed that Charles had been well onside, with team-mate Kerr in an offside position; while the Australian did make a run towards the ball, it did not seem significant enough to rule out the goal.
Hayes, who will leave Chelsea at the end of the season to take over the US women's team, was livid with the officiating afterwards and slammed both the penalty and the offside.
'Of course maybe Jessie [Fleming] shouldn't lunge in that area, but it happens clearly outside the box.
'At this level when you are in control of a game and important decisions like that go against you, it makes it difficult.'
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She continued: 'I thought we controlled the game after the first goal. Listen it was a deflection, poor to concede but not much you can do about it. Millie [Bright] goes out to block the shot and Ann [Berger] is left wrong-footed.
'I thought we dominated the game. We went 2-1 up. Then there is the free-kick that was awarded as a penalty. Then we score a really legitimate goal with Niamh Charles three or four yards onside.
'It is embarrassing. I had to check why it was disallowed; was it Niamh? No. Was it Sam [Kerr] interfering with the goalkeeper? She was about seven yards away, no way near the keeper. I cannot understand the decision whatsoever. I think we've been robbed of what was a 3-1 game.'
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