So yeah, client crashes if you alt+tab while in fullscreen mode (yeah yeah I know... I'm hoping this gets fixed)
In the meantime though, you're not completely screwed if you've got Steam (which btw is free). Register FFXIV as a non-Steam game and run it through the Steam client and you can have the Steam overlay (shift+tab in game) which gets you a rudimentary web browser. It's like a bastardized version of IE but it's better than nothing at least until this gets fixed.
In fact, this post is being made in that very browser with the game running underneath it in fullscreen mode.