Doing multipitch routes with double ropes is great, but (at least for me) inevitably results in messy tangles, twists, and crosses of the two ropes.
What are some strategies to prevent the ropes from tangling?
Definiely do not attempt to keep them separate on the route. Uncoil and stack them separately at the start, but thereafter handle them as one.
Make sure that leader and second both tie the same rope on the same side so that they are not crossed over.
I've never used a rope basket, but it doesn't seem like a bad idea if you don't mind the weight. Otherwise the belayer should normally lay the rope(s) back and forth in front of him across his tether to the anchor. (People talk here about all kinds of increasing-decreasing size loops and stuff. In reality, "reasonably tidy" is achievable). The exception is a large, flat, comfy ledge: here laying them on the floor is ok.
I nearly always climb with partners with whom I alternate leads, so the need to rearrrange the ends if the same climber is leading everything doesn't arise. If this is not your situation, the couple of minutes it takes to pull them through might well be preferable to the clusterf*ck that can result from a botched attempt to "flip the stack".
From this site:
The Rope Basket