Synthetic underlay changes that.
Strip the whole roof as fast as you can then cover it with the synthetic underlay. It's waterproof and 10,000 better than any tarp. Once it's down the roof is sealed and you can work in the rain on it. It can be left own it's own for a while, just requires a more nails with a wide cap head if being left without shingles to prevent wind tear off.
http://www.iko.com/products/homeowner/synthetic-underlayments/roofgard-cool-grey/
If you've never used it the stuff will change your mind on when and how a roof can be done.
That is the same (or similar) stuff to what they used when they did my roof since it was done in the winter.
(Feb. 2010, the week before the Olympics. The rains that washed away all the Olympic snow also finished off my roof. I slathered on enough roof sealer tar that I got the leak to stop but once we got a break between storms we thought it was a good idea to get it fixed permanently.)
I was impressed with the material, looks similar to a white tarp/Tyvek material with extra reinforcing fibers in it. There is no way you could tear it, needs a sharp knife to cut.
