Torquator Tasso is the second of seven foals out of the placed Tijuana (Toylsome), who has also produced Tunnes (Guiliani), winner of the Gr.3 Herzog von Ratibor-Rennen, and the winner Tellez (Guiliani). Tijuana is a half-sister to three stakes performers, the best of them Tusked Wings (Adlerflug), winner of the Diana-Trial, then a Group 2. The others are Titurel (Dr Fong), third in the Gr.2 Prix Maurice de Nieuil, and Tangut (Adlerflug), placed in the Gr.3 St Leger Italiano. Further half-siblings are the Listed chase winner Tahini (Medicean) in Italy and the dam of the German Listed-placed The Dancing Fairy.

Second dam Tucana (Acatenango) won twice. She is a half-sister to the sire Tertullian (Miswaki), a Gr.3 winner in France, to the German Gr.3 winner Terek (Irish River), to the Irish Listed winner Tusculum (Sadler's Wells) and to the German Listed-placed Tabari (Sadler's Wells).

Third dam Turbaine (Trempolino) was stakes-placed as runner-up in the Listed Prix Occitanie and is a half-sister to Urban Sea (Miswaki), winner of the Gr.1 Prix de l'Arc de Triomphe and dam of the multiple Group 1 winners and sires Galileo (Sadler's Wells) and Sea the Stars (Cape Cross).

Urban Sea is a daughter of Allegretta (Lombard), a full sister to Alya, the second dam of his sire Adlerflug. As a result their twice stakes-placed dam Anatevka (Espresso) appears 4x5 in Torquator Tasso's pedigree.

Tijuana has an unraced 2-year-old filly, Tiara Hilleshage (Adlerflug), a yearling colt by Adlerflug named Tiamo Hilleshage and a filly foal by Guiliani.