I think a member of the royal family has to become a 'patron' of the club before it can use the 'Royal' prefix. It's not just that a royal has played there - otherwise Prince Andrew would be creating dozens of new ones every year!
There are 34 Royal Courses in the UK. 20 in England, 8 in Scotland, 4 in Northern Ireland and 2 in Wales. It begs the question whether the Royal seal is handed out to much-with so many clubs with the title does it takes away from the honour? Maybe the rules for being given the title should be tightened up.
There are only 3 in Northern Ireland as far as I know - and I live here. There's Royal County Down, Royal Portrush and Royal Belfast. If there are any more I'd like to hear about them! There's also Royal Dublin in the Republic of course.
An interesting point Muttlee, because there are only three course’s given the name in the manor you would expect, but there is also Down Royal Park Golf Course, County Antrim. It was given the Royal name because it lies within the perimeter of Down Royal Race Course-which was granted a Royal Charter in 1684 by James II- another way of getting the Royal name.