Thanks for the suggestions, everyone - except for the umbrella one. You've clearly not been in a situation where an umbrella is zero help and you can't control your camera.
With the 6D, I can control the camera remotely (through my phone) if it's on a tripod and aimed in a fixed location, but I really need to be able to shoot on the fly (access zooming, focusing, buttons, etc. manually as I carry the camera. I also use the LCD screen, not the viewfinder.
With the one in the link I posted, the back closes up to cover the back, but then you also can't see the screen anymore.
Still considering options here...