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I'm mainly looking for a solution that punches through NAT the way Tailscale does.

Nebula has NAT punching as well[1]. Or does Tailscale do even more magic on it?

It seems to have relays which is interesting. For comparison, here is Tailscale's article on how they do NAT traversal: https://tailscale.com/blog/how-nat-traversal-works/