I've just been digging a bit more into this issue 🙂
For me, what causes the problem in the first place is command+TABing to go to another application within another space - that's the quick way for me - I usually have my apps allocated to their own space/desktop to avoid clutter. Either that or simply selecting an app in another space. You then need to actually USE the other application otherwise the issue doesn't arise. When you then command+TAB back to F360 (or select it in the dock), the browser will disappear.
However, if you use exposé (F3 or fire up the exposé app) to show all apps in the current desktop then click on the orphan browser window, it WILL come back, BUT if you click anywhere in the design window of F360, it will instantly disappear again, so this solution doesn't work permanently (for me at least :-)))
If you ctrl+right arrow away to the next desktop then ctrl+left arrow back again, the orphan browser window will return permanently - this simply switches spaces/desktops and then back again.
Alternatively, use Exposé (F3) to select another desktop from the list of displayed desktops at the top of the screen then select the desktop containing F360.
If all your apps are in the same space/desktop, then the issue doesn't arise of course - this was proposed as a 'solution' by the F360 team :-))
Update, simply activating any other app/window/desktop background within the SAME space/desktop as F360 will fix it too - that's a better 'solution' methinks :-)) That's assuming you're not in full screen mode where you can't see anything else of course LOL :-))) I guess a small applescript that does that and attached to a spare mouse button might be another quick workaround (you are all using a real Logitech mouse and not the toy Prada ones supplied by apple :-))
Cheers
Neil