the.com/petticoat government

means A household, office, or realm where women hold the real decision-making power, even though men nominally sit in the visible seats of authority.

from From "petticoat" — the voluminous underskirt that was shorthand for women and womanhood in English from the 1500s onward. By the 1700s, "petticoat" was being used adjectivally to mean "female" in a faintly mocking way (as in "petticoat influence"), and "petticoat government" followed as a jab at households or courts where wives, mistresses, or mothers were seen as the true power behind male figureheads. The phrase carried an edge of male anxiety — it was rarely a compliment when coined, even though it often described simple competence.