wash one's dirty linen in public
Also air one's dirty linen or laundry. Expose private matters to public view, especially unsavory secrets. These metaphors are reworkings of a French proverb, Il faut laver son linge sale en famille ("One should wash one's dirty linen at home"), which was quoted by Napoleon on his return from Elba (1815). It was first recorded in English in 1867.
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