How to use vim-slime from within a single tmux session?

Okay, so I was having exactly the same problem, which is what brought me to this question. I have a split session, vim code on the left and a scheme prompt on the right.

My problem was, I thought the session name was the socket name, but they are two different things. I had named the session '0', for the 0-th window, but in fact, the SOCKET is named 'default' despite the session name I specified.

To get a list of the tmux sockets run: lsof -U | grep "^tmux"
I found that from this answer:
https://stackoverflow.com/questions/11333291/is-it-possible-to-find-tmux-sockets-currently-in-use

The above was helpful to see the actual names of the sockets.

That is what you put in the first prompt. I was putting '0', which was the name of my session, but it was not working. 'default' is what is needed there despite the fact I had named the session.

Then, at the second prompt, I entered (index-0 window, index-1 pane):
:0.1

Voila! Finally! It was working. Brilliant, now side-by-side editing!