I have found that when you have Vmix running and outputting to a display and you want to do other tasks on the computer without interrupting that display, you have to be very careful that you don't inadvertently hover your cursor over an icon in the Windows taskbar because it will momentarily display your computer desktop screen instead of Vmix. Mind you, it doesn't show that you are opening up a program on the display - it shows the static Windows desktop and then jumps back to Vmix after you load the other program.

Has anyone else dealt with this problem and come up with a solution other than just to be super careful that you simply click the taskbar icons without hovering on them in the process?

I think the best way is to use an external output to an SDI output card. This requires adding another card to your computer, and hopefully you have a free slot and enough PCIe lanes to spare. But that will be the most reliable output.