It's mucus. If your eyes are dry and/or you have allergies you're going to get it. Use lots and lots of preservative free artificial tears to wash it out. (good products are Bion Tears, Thera Tears, or anything that has no preservative and doesn't claim to "get the red out")
Personally I have become an expert at removing these annoying strings from my own eyes. I wait until I can visibly see it in the corner of my eye and then just lightly touch it with a q-tip and pull it away. It comes right out and the relief is instant.
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