Are you running the game in Full-Screen mode or Windowed (Borderless/Full-Screen) mode? (In-game graphic settings). If you just have full-screen it may be worth trying Windowed (Borderless/Full-Screen) as this should allow the game window to remain open and running when you have other elements open (i.e. start menu, web browser). If I recall alt-tabbing an application in full-screen mode effectively tries to minimize it, which some games will not like. Overall it sounds like a graphics driver/directx glitch when it goes to black permanently, though I had similar issues with Radeon cards running in Freesync mode with both FFXIV and other games previously (switched to Nvidia and not had issue, but I do have it limited to 60Hz refresh rate).