Sure, I could come here to work without a male sponsor. If I'd been (officially) hired in the U.S. to come here to do this job, my sponsor would be the Qatar Foundation, the umbrella organization under which this university falls. However, because I got the job late in the game, I'm officially here as foobart's wife who coincidentally has been hired as a local. Thus, my husband is my sponsor. I couldn't have a female sponsor, because my sponsor would always either be the company that hired me or the family member who has been hired, i.e. my spouse. (And in Qatar, my spouse could never be a member of the same sex.)

For what it's worth, this is a BETTER situation than in the U.S. If we were Qatari and foobart had been invited to the U.S. on an H1B visa (which is the equivalent of what he has here), I would be granted an H4 visa, which means I wouldn't be allowed to work in the U.S. at all. So it's not like the U.S. promotes spousal equality among immigrants! (And same-sex relationships are likewise unacknowledged for visa purposes.)