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.