Rescue throw bags for water work this way, and work well. Climbing ropes, especially the 1/2" static lines used for loooong rappels are pretty stiff and not given to bad tangles anyway.

Having the rope stacked in a bag is a great way to throw the rope down a cliff for rappel when it is windy or for helicopter use. At a lot of challenge/ropes courses they use a five gallon bucket to stack the rope in, to keep it clean, portable and free running.

For carrying in a vest pocket, pick a stiff cord, like 2mm climbing perlon or levalore blind cord for a very small diameter that doesn't tangle.