Korekiyo mentions his Sister several times in his Free Time Events with both Kaede and Shuichi, mentioning that she was sickly and spent a lot of time in hospital, she made his uniform for him and that he learned anthropology thanks to her.
However, the truth behind their relationship is revealed once Korekiyo's double crime in Chapter 3 is found out. Korekiyo says that he and his Sister were lovers. But because he was her brother and lover and because of her constant visits in the hospital, she didn't have many friends. She eventually died. Remembering how lonely she used to be when she was alive, Kiyo decided to kill girls he thought were worthy of being his Sister's friends, believing that now that she's a ghost, her friends should be ghosts as well. He set a goal of making 100 friends for his dead Sister, thus becoming a serial killer.
Additionally, Korekiyo claims that during one of his seances, her spirit visited his body and has ever since been inside him. Several times during the trial, Kiyo takes off his mask, revealing he was wearing a lipstick underneath, behaves in a different manner and talks with himself. However, as Maki pointed out, this may be a case of delusions after the traumatic experience of his Sister's death rather than a case of a split personality or a ghost possession.
During his execution, despite being happy to see his brother again, Korekiyo's Sister takes part in the exorcism, along with Monokuma, to get rid of his soul. This might mean that she did not appreciate him killing girls for her.
However, with the reveal in the final chapter, it's unknown whether the relationship he had with his Sister was real or whether it was a part of his fabricated backstory by Team Danganronpa. It's even unknown whether he had a sister at all.
This ship is one of the least popular Danganronpa ships overall. The fact that it's a canon incest relationship was enough to put off some fans. The reveal that the sister's death had a negative effect on Korekiyo, with him wanting to kill 100 girls so she'd have "ghost friends" in the afterlife put off pretty much the rest of the fandom.
- KoreSister is one of only four ships to be made canon in the Danganronpa series thus far- the others being Hijirisano, Monocest, and Ruruyoi. It's significantly more common for ships to be left ambiguous or for one of the characters to die before the story ends.
- This is one of two Danganronpa's canon incest relationships, with the other being Monocest. However, as it's unknown how many of Korekiyo's memories were fabricated, this one might not count.