Movie is supposed to be an autobiography loosely based on their lives. They got married in 2007, there is a scene in which Kumail uses Emily's fingerprint to unlock her phone, but that technology was not present on the iPhones then.
Early in the film we see Kumail throw a photograph on the pile of pictures in his cigar box. Below several other pictures is a photograph of a woman in a light blue hijab. Later in the film he again throws a photograph on the pile. It is the same picture of the woman in the light blue hijab.
Near the end of the movie when Kumail is looking through the window in the hospital room door he can see Emily's head but when the shot switches to the other side of the bed it is obvious you would not be able to see her from the window because of the angle of the bathroom wall.
Although the film is set in Chicago and Emily is said to be at "City Hospital," a sign for Katz Women's Hospital at the Long Island Jewish Medical Center in New York is clearly visible when Kumail arrives.
When Holly Hunter is signing papers to discharge her daughter from the hospital, she checks the time on her watch by looking at the bottom of her wrist. However, the face of the watch is on the top of her wrist, so she wouldn't be able to see the time.