More environment & testing information
My host OS is Mac OS X 10.9.1 running on a mid-2009 13" MacBook Pro, 2.53GHz Core 2 Duo, 8GB memory, NVIDIA GeForce 9400M 256MB video, 512GB SSD system drive.
I have an external monitor (1920x1080 Hannspree flatscreen) connected via Thunderbolt-to-VGA adapter. It is configured as the secondary display, and the "Displays have separate Spaces" option is enabled in Mission Control.
I have several guest OSes, including Win7, WinXP, Linux Mint 15, and tails.
I can run any guest OS in fullscreen mode on the built-in display with no problems.
I can run any guest OS in windowed mode on the secondary display with no problems.
If I run any guest OS in fullscreen mode on the secondary display, the display will go completely grey at some point. Sometimes it takes a few minutes, but it always happens.
The "tails" guest OS is a CentOS-based VM that does not have Parallels Tools installed, and the grey screen issue occurs with it as well.
There doesn't seem to be any "magic action" that causes the greyscreen, but in general, it seems like the more video activity, the faster it will occur.
When the screen goes grey, I can use Mission Control to drag the VM's screen from the secondary display to the built-in display, at which point the video returns and I can continue to use the VM. The grey screen does not appear to affect the operation of the VM, but only prevents me from seeing the VM's screen (i.e. the guest OS is still running "behind" the grey screen).
So, based on my experience:
* The issue is not with Parallels Tools, because the greyscreen occurs even when the guest OS is not running Parallels Tools.
* The issue is not with any particular guest OS, because the greyscreen occurs with multiple different guest OSes
* The issue is not with the video settings of the guest OS, because I've tried multiple configurations with multiple guest OSes
It appears to me that Parallels Desktop itself has a problem in fullscreen mode with secondary displays in Mavericks, at least on some systems. No clue what's happening there.