I've personally played with Gemini a few months ago, and now want a new Internet as opposed to a new Web.

Replace IP protocols with something better. With some kind of relative addressing, and delay-tolerant synchronization being preferred to real-time connections between two computers. So that there were no permanent global addresses at all, and no centralized DNS.

With the main "Web" over that being just replicated posts with tags hyperlinked by IDs, with IDs determined by content. Structured, like semantic web, so that a program could easily use such a post as directory of other posts or a source of text or retrieve binary content.

With user identities being a kind of post content, and post authorship being too a kind of post content or maybe tag content, cryptographically signed.

Except that would require to resolve post dependencies and retrieve them too with some depth limit, not just the post one currently opens, because, if it'd be like with bittorrent, half the hyperlinks in found posts would soon become dead, and also user identities would possibly soon become dead, making authorship check impossible.

And posts (suppose even sites of that flatweb) being found by tags, maybe by author tag, maybe by some "channel" tag, maybe by "name" tag, one can imagine plenty of things.

The main thing is to replace "clients connecting to a service" with "persons operating on messages replicated on the network", with networked computers sharing data like echo or ripples on the water. In what would be the general application layer for such a system.

OK, this is very complex to do and probably stupid.

It's also not exactly the same level as IP protocols, so this can work over the Internet, just like the Internet worked just fine, for some people, over packet radio and UUCP or FTN email gates and copper landlines. Just for the Internet to be the main layer in terms of which we find services, on the IP protocols, TCP, UDP, ICMP, all that, and various ones and DNS on application layer, - that I consider wrong, it's too hierarchical. So it's not a "replacement".