I can understand allowing people to verify their identity, but I cannot fathom being a right winger and being against anonymity on the internet in the current situation.
I do not desire mandatory real names everywhere on the internet, what I want is for it to be the norm in many venues. It would do a lot to mitigate astroturfing. Shareblue and act.il shills and hirelings would be exposed just as much as we would be, and in the long run that would hurt them far more than us. Anonymous venues would have the advantage of anonymity... and the current disadvantage that 99% of current sites have of being easily swamped in bullshit when moneyed interests decide to do so.
As for the political aspect... the use of screen names provides only an illusion of safety. With modern electronic fingerprinting techniques, leftists employed by Google/Amazon/Facebook/etc have all the information they need to dox 99.9% of everyone. It would not surprise me if "anonymous" doesn't have to hack anything, but uses the claims of hacking to hide their use of corporate information gathering.
The illusion of safety does not protect anyone who actually does something that matters, but it does allow the internet to be used as a safety valve where people can expend their will to act fruitlessly... and to identify themselves for later harassment when the Overton window is shifted.