I'm not sure what the concept of "tab" is in your head; however, from what you want, windows are more than enough.
Imagine that you have a “tab” in tmux, then you more or less need to switch to this tab to see the result, so the operation is exactly the same as in the tmux window.
Usually I type <cx> c to have a new window, and then return to the main workspace (previous window) with <cx> p and open my vim there to work. I could have several panels in my main workspace using the tmux panel (not with the vim area, because I need the terminal to always orient). Then, after I changed everything, save, just enter another <cx> p to go to the server and see what happens there.
(note that I am binding <cb> to <cx> )
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