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tmux + ttyd (or gotty) could be used in a similar way. But it's not like an extra screen, it's more like mirroring a terminal to another device.

Someone asks for a webpage and not only do you propose tmux, you also acknowledge it doesn't even do anything like what's asked.

Typical Linux user.

Gotty serves a terminal window as a website. Tmux can be used to force the terminal size, so that you can a small window for typing, like OP does.

The person on the other end does not have to learn either of them.