Akaki Tsuki No Meguru Koro

Riu Rokugiya is the only daughter of a famous Kabuki theatre house. Since she was small, she had always dreamt the same dream about meeting a man under the red moon. A man she didn’t know, until she attended one of the performances of her brothers. The man turns out to be one of the Akatsukiya’s heir, Rui.
The Akatsukiyas were distant relative of the Rokugiyas. Just recently, its Head, Ashikawa Enjuuro died. The family is on the process of choosing its successor among the three adopted sons, one of which is Rui.
What relation will Rui have with Riu? Why was she dreaming of him dying every night?