Actually a breakup can turn into any one of those situations. One person can begin spreading rumors and propagating lies about the other person, or begin leaving harassing messages. Workplace morale can be severely affected. Co-workers lives can literally be in danger because of how one person in a workplace romance interprets behaviors.
You really have to understand the perspective of a company to know why any form of singling out an employee for any type of reason, including dating or romance, is a potential legal landmine.