I assume you've already perfect the art of tying knots in Remay when it rips?
I think maybe you need to change your attitude. Don't get me wrong, because I know using remay can suck a big one. You didn't even mention all of the time + capital expenditure that goes into holding Remay down on the ground and trying to fight loose remay on a windy day (maybe that's just an Illinois problem

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Maybe it you approach this issue knowing fully well that Remay sucks and is expendable then you'll be a little more relaxed about the process. First off you need to have at least three separate and distinct storage areas for remay. 1) New remay on rolls stored in a dry place away from rodents like rafters of the tool shed 2) Used remay that is in good condition and can be reused for bug and/or frost protection 3) Remay that has been used and is significantly torn but can still be used for things that need overwinter protection from the elements such as garlic, strawberries, spinach etc. You will probably use a double layer in these situations 4) Dumpster or Upcycle