Tate's international conducting début was with the Metropolitan Opera in New York City in 1979. In 1985, he was appointed the first principal conductor of the English Chamber Orchestra. He was named to the position of principal conductor of the Royal Opera House, Covent Garden effective in September 1986, the first person in the House's history to have that title. He was principal conductor of the Rotterdam Philharmonic Orchestra from 1991 to 1995. In 2005, he was appointed music director of the San Carlo Theatre of Naples, and served in the post through 2010.
Tate was openly gay. He was partners with German doctor Klaus Kuhlemann until his death in 2017. Tate was created a Knight Bachelor in the 2017 New Year Honours for services to British music overseas.
- David Blum (19 June 1994). "Bucking the Biggest Odds of All". New York Times. Retrieved 2007-10-12.
- "Royal Opera Appoints Tate as Top Conductor". New York Times. 6 December 1984. Retrieved 2007-10-12.
- David Blum, Quintet: Five Journeys Towards Musical Fulfillment, p. 59. Retrieved 11 January 2014
- London Gazette: , 31 December 2016.
- Sad news: Eminent conductor collapses and dies