When Sting used the word to describe his passionate ...
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In an interview in Harper's Bazaar, the musician Sting used the word to describe his passionate marriage: "Romantic? We like tawdry."
Tawdry means "morally low or bad," or "cheap or gaudy in appearance or quality."
Sting used it here as a slightly more refined alternative to a word like "sleazy."
As it happens, the word's origins are romantic. Its initial meaning of "cheap or gaudy" comes from tawdry lace, which was a tie of lace for the neck, which comes from the sound of "St. Audrey" - an ancient queen of Northumbria (a kingdom in what is now England), who presumably wore that kind of lace.