The problem: you have two vim windows opened side-by-side in two tmux panes, and your vim configuration does not allow copying into the clipboard.
The solution: select text to copy using the Visual mode in vim, then :w buffer_file; in the other vim window, :r buffer_file where you want to copy. Here’s the situation. I often use vim when working on a remote machine. I also use tmux to split my wide display into parallel panes, which is extremely useful for editing one file while referencing another. The first approach is to enable the copy into clipboard mode of vim, but I have no control over the vim configuration. The second approach is to copy using normal copy technique on the console (i.e. highlight text and then right-click / copy or Ctrl/Cmd C), but this does not work when you have two tmux panes running side by side. Hence the above solution.
Comment: This solution is so lame now that I know that Vim also support splitting a window. Personally I use NerdTree but just using the built-in feature of Vim should work too.