In Sweden, Chelsea could not turn control into a goal. They had 20 shots to Häcken's three and 76% of possession, and they were still level with a couple of minutes of stoppage time remaining. The match finished 0-0.
Real Madrid's win came in the second half. Arianna Caruso scored in the 55th minute and Lineth Beerensteyn in the 65th to open a 2-0 lead, before Caicedo added a third in the 81st minute. The final score was 3-0.
The evening game is scheduled for an 8pm GMT kick-off. City begin this set of fixtures ninth in the 18-team league phase, with only the top four guaranteed a place in the last eight.
After drawing their first three games, City need to start winning if they are to progress automatically to the quarter-finals.
Inter, in the competition for only the second time, finished second in Serie A last season. The match is another chance for them to test themselves against the best.
They have already shown they can produce an upset, having eliminated two-time champions Wolfsburg in qualifying in September.
The draw meant Chelsea missed the chance to go top of the 18-team league phase. Leaders Barcelona host Arsenal on Thursday.
Chelsea's Lucy Bronze set a record with her ninth appearance in the world XI. Manchester City's right-back was not included, though she was the fifth-highest defender in the Ballon d'Or rankings and 24th overall.
City's task against Inter is straightforward on the table: three draws have kept them in the race, but only wins will secure an automatic place in the last eight.