We have just had a similar case where I work. The couple in question were taken to the manager's office and given a disciplinary hearing. At the hearing they argued that their personal life was nothing to do with the company and that as they worked in different offices, used their break times to meet and converse, and the affair wasn't affecting the company or anyone connected with it, there was nothing the company could do. They have the same contract of employment as I do, and there is only a tiny section that says 'any action likely to bring the company into disrepute' that would seem to be appropriate. As they aren't openly showing affection (i.e. snogging or shagging) on the premises and they have no contact with the customer base, it would seem that there is no reason for the company to get involved. However, the manager had to be seen to act as 'someone' (we know who it was and she is a mindless busybody!) complained. Eventually a Director stepped in and put a stop to the whole thing, getting written agreement from the couple that they would not contact each other unless it was work related or they were off the company's property.
How wrong is that!! And all because some shrew was jealous of their happiness.