Triple Time is one of nine winners out of Reem Three (Mark of Esteem), a three-time winner who was Listed-placed, and he is her third Royal Ascot winner and one of six black-type winners from her. The others are Ajman Princess (Teofilo), winner of the Gr.1 Prix Jean Romanet, Ostilio (New Approach), winner of the Britannia Handicap and the Gr.2 Prix Daniel Wildenstein, Cape Byron (Shamardal), winner of the Wokingham Handicap and the Gr.3 Bengough S. and placed in the Gr.2 City of York S., Third Realm (Sea The Stars), a dual Listed winner whose wins include the Listed Lingfield Derby Trial and who was Group 3-placed, and Captain Winters (Lope De Vega), winner of the Listed Heron S. Imperial Charm (Dubawi), a winner at two, was third in the Prix Saint Alary, then a Group 1. Reem Three is a half-sister to Afsare (Dubawi), winner of the Gr.2 Celebration Mile and the Gr.3 Sovereign S., twice Group 1-placed and second in the Grade 1 Arlington Million.

Second dam Jumaireyah (Fairy King) won twice. She is a half-sister to Lost Soldier Three (Barathea), a Listed winner who was Group 3-placed, and to Zeeba (Barathea), a winner at three and dam of Danadana, winner of the Gr.3 Huxley S. She is also a full-sister to the Listed-placed Altaweelah, whose daughter Qasirah was Listed-placed too.

Third dam Donya (Mill Reef) was placed and is a half-sister to French Glory (Sadler's Wells), winner of the Gr.1 International S. and later a sire.

Reem Three has an unraced 2-year-old filly by Kingman named Bolsena and a yearling filly by Night Of Thunder.