Whenever I start a fullscreen window (using SFML-2) the 2nd monitor turns itself off. This happens in both Windows and Linux.
Is there a way to open a fullscreen window without this happening?
Well it doesn't happen for me. I'd guess it's your graphics card/driver that does this.
What graphics card do you have and is the driver uptodate?
Can you run other games in fullscreen with the 2nd monitor not turning off?
Or is there any way to make a "normal" window lie on top and cover the entire screen, instead, in effect a "fake fullscreen"?
Yes you can do that too. Just set the window style to I think
sf::Style::None and the resolution to
sf::VideoMode::getDesktopMode().
Keep in mind though, that it will lead to a small FPS drop.