You could always solve these nomenclatural issues by elevating the Principality to a Kingdom, which would then be a part of the United Kingdom. 
Wales isn't actually a Principality, is it? Obviously there's a Prince of Wales, but he doesn't actually exercise any sovereignty in Wales.
he has some ancient feudal perks that the queen has in England(except Cornwall of course, where Chaz is the Duke) the only one I can remember off the top of my head is that he gets first dibs on any Sturgeon caught in Welsh or Cornish rivers
