Anonymity isn't a problem when everyone's anonymous. Privately harassing an anonymous person accomplishes little to nothing, and generates no satisfaction for the harasser, because the victim is free to change their pseudonym, and their real life isn't affected, and the harasser doesn't get to witness the results of their behaviour. And if an anonymous person wants to harass a person publicly under their real name, they'll have to do it in a public forum, where the content can and should be moderated. This (moderation) is where South Korea seems to be failing.
Hold the chat rooms etc. legally responsible for what users say there, and they'll start actively moderating and banning abusers - presto, problem solved while keeping anonymity intact.
Anonymity becomes a problem when the 'bad guys' have the choice to be anonymous while the 'good guys' are readily identifiable. That's where you get serious harassment that spills out into real life, embarrasses people, hurts their employment prospects, etc. And that's exactly the situation you get when you 'require' the use of real names, because it's simply unenforceable.