On the responsive design on a mobile device (e.g. iPhone), the account popup menu fills the screen, so there is no way to get rid of it.
How hard would it be to add a close link (e.g. a font awesome circled X button) located top right of the popup box to close it?
It would likely require some custom JS to trigger the menu to close (or proxy click outside of the menu).
That said, your menu is, for some reason, somewhat wider than the default and I can't particularly see a good reason for it to be. A simpler solution may well be to reduce its width. On a 4 inch phone screen, there's still plenty enough space to tap out of the menu on the default style.