What happens when you plug a PlayStation 4's HDMI cable into the back of an Xbox One? The same thing that happens to everything else.
The fine folks at DigitalFoundry plugged their PlayStation 4 into the Xbox One's HDMI input and — contrary to my fevered speculation — a singularity was not formed. Instead, they got the PS4 video in their Xbox One dashboard. From there they could navigate PlayStation menus, fiddle about with settings, or even launch a game, though lag issues make it a less-than-ideal means of doing so.
Either way, it's lovely to see the two of them getting along so well.