I explained it poorly in my comment above. The mind-reading analogy is useful, but it's just an analogy. Otherwise the solution would be "Use your amazing psionic powers to level both enemy cities without leaving your room".

If I had to extend the analogy, it might be something like this: we take a pair of strategies and run two checks on it. The first check is "If your opponent's choice was fixed, and you alone had mind-reading powers, would you change your choice, knowing your opponent's?". The second check, performed in a different reality unbeknownst to you, is "If your choice was fixed, and your opponent alone had mind-reading powers, would she change her choice, knowing yours?" If the answer to both checks is "no", then you're at Nash equilibrium. You don't get to use your mind-reading powers for two-way communication.

You can do something like what you described - if you and your girlfriend realize you're playing the game above and both share the same payoff matrix, then (go home, go home) is the obvious Schelling point because it's a just plain better option, and if you have good models of each others' minds you can get there. But both that and (stay, stay) are Nash equilibria.