Answer: It will be quite a challenge for the runners to take root on the plastic. You can place a thick layer of straw over the plastic so the runners can root, then cut the new little plants and transplant them after they've developed roots. Or, cut slits in the plastic wherever the new plants develop, to allow them to root directly in the soil. Strawberry plants should be renewed every 3-4 years, and the runners can produce replacements for the parent plants. If you cut the runners off, you won't have renewal plants for your beds, but it won't hurt the parent plants. The choice is yours.
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