Large organisations such as RBS are very very careful about their reps these days, especially since the CEO gets his name dragged into a Google search. I suspect at the very least someone may see it, and someone will have a word with the named employee.
The last large such organisation I worked for, before the one I'm now at, was incredibly worried about bad publicity, and bringing the company's name into disrepute (even if it appears minor, this is the type of thing where mud can stick, especially in this industry) could lead to disciplinary action.
"What employees do in their spare time is their own business, surely."
Ah if only. For reference note people who have lost their jobs after tweeting about hating cyclists and wanting to kill them. Entirely in their own time. Entirely done away from work. Their workplace never mentioned. Jon Ronson has written a book on social media shaming - it can be remarkably destructive.