How can they route without an IP address? Are you referring to a cgnat?
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what do you mean how can you route without an IP address? they have one, internally, in the ISP's WLAN, they don't have one that can receive inbound connections.
you don't need an IP address on the internet to make connections out, CONNTRACK on the router handles the internal routing without that.
how it looks is the user has an IP address in common with dozens, maybe hundreds or more other users on the same last mile route. this can be in front of a mobile network as equally as a wired network.
it's getting less common but i'm sure there is still shitty ISPs that don't own enough IP addresses to give all of their customers their own.
from a technical standpoint, there is several layers to this
ARP is how the routers talk at the base layer
IP is a secondary layer that provides routing capability
having a private IP on a non-routeable address range can still make outbound connections through a router that does have a public routable internet IP address.
how it looks internally, if you can connect a PC directly to the outbound-only capable network is your ETH device shows it has no address, but it has a MAC address which is how ARP from the router's PHY can find the path back to send you the responses to your requests
i also should point out that a lot of cheap domestic home wifi routers have "wifi isolation" on by default which means you can't make connections between your devices both connected to the same wifi, without the additional use of a VPN to do rendezvous of your connection for you