Charlotte National, located in suburban Charlotte, is a privately-owned public course. I have not played there in 10 years or so, but it's not a bad course. Here's a link to something about it -- not an official website. I'm not sure if they have their own website. http://www.golfholes.com/nc/_charnat/geninfo.htm
Never played there, but one of my old courses, where I was a member, was owned by the part owner of Charlotte National. Never managed to play there though; this was before I traveled around as much as I do now.
North Carolina is a pretty big violator of the 'national' rule: Charlotte National, Carolina National, Cape Fear National, Greensboro National, Sapphire National, Sequoyah National, and The National (former Pinehurst National). And they are all public.
Virginia has its share as well: Chantilly National, Fairfax National, Reston National, Trump National, Virginia Beach National, Virginia National, and Williamsburg National. And of course International Town and Country Club. Of these, 3 are private, and of that, probably only Trump has a real 'national' membership.
One thing that can be noticed about these courses though is that while they may not be great, they are all generally fairly good, at least from my experience. I should say that in general, if you were to have one round in an area, with no other information, going to a 'National' course would not be horrible, you probably wouldn't be going to a real stinker.