So they not relied on the global general publiick internet.
Bitcoin wallets have offline "signing" capabilities. But they can not create a transaction, broadcast it to the rest of the network or sync their chain with the network without the internet.
The independent network could be some boxes around the globe wich working with radio-signale completely independent of the general global internet
What we need is a network that already exists and is used and doesn't need any special equipment. This is why the Internet works well, we already use it and have devices to connect to the internet. So bitcoin could be built on top of it. An alternative should be similar.