The phrase "major embarrassment" refers to a situation or event that causes someone to feel very ashamed, humiliated, or awkward. It usually involves making a mistake or behaving in a way that others find embarrassing. Full definition
When his findings appeared in Nature, they caused a sensation — and major embarrassment at NASA. (newscientist.com)
The black ankle pants from Club Monaco were a must but the cream sweatshirt I paired with it was ill - fitting which she pointed out preventing major embarrassment and the loss of substantial cash. (40plusstyle.com)
Any glitches, and it could be a major embarrassment for either company. (inc.com)