The BBC said the West End show, which moves William Shakespeare's comedy from 16th century Sicily to 1980s Gibraltar, opened this week to largely good reviews.
Tennant and Tate play sparring partners-turned-lovers Beatrice and Benedick in the production at Wyndham's Theatre, the BBC said.
The show is set to run through Sept. 3.
Tennant played the Doctor on "Doctor Who" from 2005 to 2010, while Tate played Donna Noble from 2006-07.
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