Middle English besme, from Old English besma "bundle of twigs" (used as a broom or a flail), from West Germanic *besman- (source also of Old Frisian besma "rod, birch," Old Saxon besmo, Old High German besmo "broom, besom," German Besen, Dutch bezem), which is of unknown origin, possibly from a non-IE substrate language.

"of or pertaining to a wife," 1778, from Latin uxoris (see uxorious) + -al (1). Sometimes is used in the sense of uxorius.

We still say that a husband hangs out the broom when his wife goes from home for a short time; and on such occasions a broom besom has been exhibited as a signal that the house was freed from uxorial restraint, and where the master might be considered as a temporary bachelor. [Samuel Johnson and George Steevens, notes to "The Tempest," 1778]