A marriage for appearances sake, between a homosexual and a heterosexual partner.
Usually the female of the marriage, was the heterosexual, and was referred to as 'the beard'
Coined mainly for when Hollywood notables, perceived as being gay, became hitched, to quell the speculation that they might be homosexuals.
Rock Hudson was one such star, that had been perceived to have entered into a marriage of convenience, to attempt to mitigate the rumours circulating at the time.
The phrase was made popular during the 1920's when the studios brought in morality clauses, to protect themselves.