Perfect Blue/WMG
The "real" Mima was a mental specter trying to possess Mima.
Like Dom's projection of Mal, she formed out of Mima's issues with leaving the idol group. When Mima became an actress, she was having a hard time staying the dominant personality, so she had to find a way to retake Mima's mind. But during the course of the movie, she's forced further and further out of Mima's mind by other issues until she ends up possessing Rumi instead. The first few times "real" Mima appears, she's in Mima's reflection. After that, she appears in other places entirely. When Rumi was trying to kill Mima, we still saw her reflection, even though "real" Mima was chasing Mima down the street. And at the end of the movie, when Rumi looks in the mirror, her reflection is "real" Mima, signifying she's been completely taken over.
"Real" Mima is Mima's Shadow.
She was Mima's denial that her actions had corrupted her Contractual Purity. And of course, during the climax, Mima denies that they're the same person, leading to "real" Mima trying to kill her.
The pop-star Mima and the detective-drama Mima are different people.
The detective drama Mima we saw Mima star in was actually filmed after the events of the movie, in a case of Ripped from the Headlines. The first Mima, an idol-turned-actress, was the one involved with the Me-Mania murders. However, the Mima shown during the filmed scenes was an actress in a Show Within a Show (er...within a show?) That explains why her scenes of being hallucinated and stalked were juxtaposed with Mima's hallucinations and being stalked. The first Mima did survive and get a happy ending, but the ending was so suddenly uplifting is because that scene was from the television series.
Mima killed the photographer.
Late in the movie, Mima reads Mima's Room and it mentions her going shopping. Mima doesn't remember going shopping. However, if you pay attention, it does briefly show a photo on Mima's Room of Mima going shopping. My explanation is thus: Rumi took Mima shopping to try and cheer her up. While there, Rumi took a photo of Mima - it doesn't matter why. When Rumi gets home and changes into the Mima-persona, she writes the story on Mima's Room and posts the photo. Rumi-Mima is confused enough to genuinely believe it's a photo of her. Meanwhile, Mima suffers a breakdown causing her to forget the whole incident, much like how she forgot that Rumi had come to visit her and mistook it for a dream. She then reads Mima's Room, panics, breaks down and changes into the clothes, before going and murdering the photographer. She comes home, bundles the clothes away in the bag, gets into bed and wakes up the next morning, confusing it for a dream. The M.O. is close enough to that of the previous murders that people immediately assume it's the same killer.
Rumi had psychic powers
One of the most accepted theories behind the plot of Paranoia Agent is that the main character had psychic powers that affected reality. It goes with Kon's use of Magical Realism and it plays with the way the supernatural is portrayed in movies: we always see the story from the viewpoint of the person with powers, but what is it like for the people that surround them? I think this is a good explanation behind the whole Rumi thing: Rumi somehow obtained psychic powers and the whole plot was a trap for her to ensnare Mima's psyche. At the end she fails due to her own lack of control, and Mima becomes completeley sane because she is no longer in her influence, and has grown from the experience. Unlike Rumi, who is unable to advance from the past.