I have two monitors plugged into some generic docking station, and I keep the lid open below those.
Found this from a post here on HN a few weeks ago, but apparently manufacturers of monitors don't bother to give the monitors unique machine ids, except in batches. Since everyone's two monitors are almost always from the same batch, they can't be told apart on a hardware level. I thought I had struck paydirt in that HN post (a tool that helped with this), but since the two monitors of mine aren't plugged directly into the Mac, the tool can't fix it for me. It would have to somehow work on the level of the docking station. If I didn't have bad luck, wouldn't have any at all?
But it's so much worse than you describe. In theory, I should be able to hit F3, see the little "desktops" at the top, and drag the one from the left monitor over to the right, and then vice versa. But somehow, there's always one virtual desktop that can't be dragged in this manner, and it's always the one with the applications.