Tunnes is out of the placed Tijuana (Toylsome), which makes him a half-brother to Torquator Tasso (Adlerflug), a multiple Group 1 winner in Germany, a Group 1 winner in France and Group 1-placed in England. Torquator Tasso won the Gr.1 Prix de l'Arc de Triomphe, the Gr.1 Grosser Preis von Berlin and the Gr.1 Grosser Preis von Baden.

Second dam Tucana (Acatenango) won twice and is the dam of four black-type performers besides Tijuana, among them Tusked Wings (Adlerflug), winner of the Diana-Trial, then a Group 2, Titurel (Dr Fong), third in the Gr.2 Prix Maurice de Nieuil, Tangut (Adlerflug), second in the Gr.3 St Leger Italiano, and the Listed-winning chaser Tahini (Medicean). Her winning daughter The Fairy (Night Shift) is the dam of the Listed-placed winner The Dancing Fairy.

Third dam Turbaine (Trempolino) was a listed-placed winner and produced Tertullian (Miswaki), winner of the Gr.3 Prix de la Porte Maillot and the sire of Guiliani, so Tunnes is inbred 3x3 to her. Her other offspring include Terek (Irish River), winner of the Gr.3 Preis der Deutschen Einheit, Tusculum (Sadler's Wells), winner of the Listed Ballycullen Stakes, now the Gr.3 Irish St Leger Trial Stakes, and the Listed-placed Tabari (Sadler's Wells).

This is the family of the Gr.1 Prix de l'Arc de Triomphe winner and blue hen Urban Sea and of the Gr.1 2000 Guineas winner King's Best, both half-siblings to Turbaine.

Tijuana has an unraced two-year-old filly Tiara Hilleshage, a full sister to Torquator Tasso, a yearling colt Tiamo Hilleshage, also a full brother to Torquator Tasso, and a filly foal who is a full sister to Tunnes.