Close, but not enough. I want to open the same editor session spread across multiple windows on multiple monitors.

I saw this thread, and I promised myself I wasn't going to plug Kakoune -- I think NeoVim is a great project and I want it to succeed. They deserve every bit of the praise they're getting. But I'm going to have to break my promise, because Kakoune has the exact feature you're after, and I'm using it right now. That, along with familiar keybindings, selection-oriented modal interface, tmux integration... I could go on. Kakoune is a modern modal text editor for your terminal, and it's worth trying!

Caveats: if you depend on plugins to give you an IDE-like experience in Vim, kak is not for you. Your vim plugins won't work. If you use Windows but don't want to use Cygwin, kak is not for you.

Kakoune is always super tempting to me but the plugin thing is a deal breaker for me. Not for IDE-like features but because limiting myself to vanilla vim motions feels like using a bad vim plugin by now.

At least surround.vim, targets.vim and a couple custom text objects.

I do depend on a completion plugin, other than that it might just work. I'll have a look at it - thanks for the suggestion! Did not know it existed.

Can't you do that with tmux?

No, Vim can only have one buffer active at any given time.