A new Internet would have to be governed. It would not be public, like the current one is.
Then what's the point of being anonymous on it?
Way to show your communist colors, Stalin!
It wouldn't be anonymous. You would need a license, and to conform to strict agreements (especially involving things like virus checking, firewalls). The license would be revoked as soon as you started spamming, sending out viruses, etc.
It would be barely anything like the current Internet.