Can't use AirPlay to stream video to TV and use Laptop at the same time

[antoniolafortezza]

I'm streaming a full-screen video (in Safari) from my MacBook Pro to my TV via Airplay using the 'use as separate display' function, however when I want to use my Macbook Pro's screen at the same time (to surf the web for example) the full-screen video on my TV goes out of full-screen mode. How can I fix this?

I took a video which demonstrates my problem clearer:

 

[Boyd01]

I don't think Safari can use fullscreen on one device and separate windows on another, or at least I don't know how to do it either. I am able to run other applications on the built-in laptop screen at the same time as full screen Safari on the Apple TV though.

Maybe you could use a different browser, like Chrome or Firefox on the other screen?