Fullscreen video (YouTube, etc.) keeps defaulting to the wrong display

I've recently abandoned Chrome and switched back to Safari, and for the most part all's good (apart from Backspace not being the Back button -- I'm sure I'll get the hang of it eventually!), but this is bugging me:

I have an external monitor (well, a TV) hooked up to my MBP, and it's set as the primary display. I often have a browser in fullscreen mode on both screens, and if I have a video playing on the primary display (YouTube, Vimeo, Dailymotion, doesn't seem to matter) and hit the fullscreen button in the player, about half the time the video opens fullscreen on the secondary display and not the one the browser is on. If I shrink it and go fullscreen again it appears where it should, but the inconsistent behaviour is annoying. I'm guessing it's related to Safari creating a new screen to play video in rather than simply taking over the existing screen -- is there a setting to change that?