Hi Fargo,
They're likely just mucus nets: adhesive feeding devices.
If they trail from the coral's mouth, then it is reeling in its catch.
If they are merely draped over the coral, then it could be anything from Vermetid gastropods to other sessile critters doing their own mucus netting, and dragging in their catch over an Elegance sitting in the way.
No biggie, as long as the Elegance shows no signs of irritation --which can occur when it is a neighboring coral doing the netting-and-dragging: a coral's mucus may occasionally carry stinging cells.
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Cogum manent
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