I'm confused. If the tarp is supposed to keep rain off of the playing field, and then you fold up the puddle-filled tarp as described, all the water goes onto the field. Right? And why doesn't the wet tarp, folded and rolled up, mildew?

My guess: because of how the grounds crew runs one edge of the tarp to its opposite edge, most of the water drains off into the outfield grass, where I suspect there's better drainage. It's pretty entertaining process, at least at Fenway. Just imagine this process in reverse:

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I don't know about the mildew, but my understanding is that MLB grounds crews are pretty fanatical about taking care of their equipment (if you want a career in groundskeeping, MLB is about as good as it gets. Well, pro golf would be up there, too, I suppose)