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?
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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
This system works well on stances where you are standing comfortably on a ledge, but you don't have enough space to flake the rope. It is also much easier if you are swinging leads. Your personal anchor system or tie-in is connected to your anchor, making a straight line to the rock. As your second climbs upward, you will drape the rope over your tie-in in neat, equal length coils. The coils shouldn't touch the ground; we have found that they tend to get tangled and twisted if they do. Your coils should start at your harness and move one by one toward your anchor. If you run out of space, place the next coil at your harness and repeat. This will keep them neat and less likely to tangle. The more care you take in making the coils neat, the same length and in the correct order, the less likely you are to have a tangle.
When your second reaches the belay, keep your stance as much as possible to keep your rope orderly. Once you have finished the transition, you can belay normally, giving rope from your rope basket as necessary. It also keeps the rope close at hand, so if there are tangles you can fix them easily.