Officially, the US supports the “unity government” in Libya, whose territory includes a single naval base in Tripoli and a few nearby buildings. In practice, the US is seen backing the two other extent would-be governments just a seriously.
That’s becoming more and more obvious as the Pentagon is increasingly open about its ongoing ground operations in Libya, where they are “meeting a variety of Libyans.” Less important that the sight-seeing is that the US has established two bases, one in Misrata and one near Benghazi.
You’ll notice that neither of those places is right next to the naval base in Tripoli. Rather, the Misrata base is in territory held by the Tripoli parliament, a rival government, and the site near Benghazi is in the sphere of the Tobruk parliament, another “UN-backed government.”
US officials have been increasingly frank about their desire to put more ground troops in Libya, conditioning it on a request from the “Libyan government.” The existing operations suggest that they aren’t particularly selective about which government that ultimately is.