I forget if I knew that there was an actual tune by the name Virginia Reel. Mostly when we hear that the dancers want to do the Virginia Reel we make sure that we have a medley to play for them because it can go on for a very long time.
I'm at work and lunch, so I only have access to my crummy aluminum banjo, but here is a CH version I recorded after listening to an excellent rendition on BHO as a Tune of The Week. The recording is a bit noisy because I'm in a room next to an air-handling unit. I finally did it right in the third repetition, but unwisely I went for a fourth one.