The only thing that removing anonymity would do is make it easier to harass people with dissenting opinions. Professional bad actors can switch to posting under "real people" names, just as spammers now post from home IP proxies.
I don't buy favourite argument of the US gun lobby - that only criminals ( yes by definition ) would have guns/anonymous accounts if you banned it therefore we shouldn't do anything.
You could apply that to anything that's illegal - by definition only criminals are outside the law - so why any laws at all?
I'd also be concerned about repressive governments - but I think you could distinguish between mass/public communication and private 1:1 communication. Just like in the real world there is a whole world of difference between saying something in private and publishing something in a national newspaper.
Filing a civil suit can be pretty expensive if you want a lawyer -- which, yes, you do effectively need one.
This is effectively a tax on the victims of harassment.
Obviously I don't know which legal system you are talking about - but it does raise the issue of cross-boundary jurisdiction which I think is more of a potential challenge.