"It is alleged that in or around the 66th minute of the fixture, the club failed to ensure that its players conducted themselves in an orderly fashion," the FA said in a statement.
Middlesbrough have until May 5 to respond to the charge.
City striker Sergio Aguero scored the resulting penalty to level the game 1-1, with the visitors coming from behind again to secure a 2-2 draw after an 85th minute Gabriel Jesus strike.
The result left 19th-placed Middlesbrough six points adrift of safety with three games to play.
(Reporting by Hardik Vyas in Bengaluru)
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