That being said, Redis is not really built to have clients talk to it directly. In particular it doesn't have the security features you'd need.
You'll likely want to build a simple chat server on the backend using something like node.js or
socket.io. Your clients would open up a socket communication with that server, and that server would in turn be the one talking to redis.
This looks like a great blog post of someone building something just like this.
You'd build something very similar on the backend, but rather than having a webpage to your backend using javascript over sockets, you'd write something in unity that does the socket communication. I'm not sure if this library is still up to date or works on mobile, but it might work out of the box
https://www.assetstore.unity3d.com/en/#!/content/21721 if you want to stick with
socket.io style sockets.