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Thank you murraymints for an interesting description of your way of tackling it. Having read the RHS description I am now pretty certain it is red spider mite (as, frankly, I had already suspected). It definitely started on the peach tree and went over to the nearest vine branch - the vine is enormous and the rest of it is perfectly healthy, hence my inclination to cut off the affected branch. By all accounts this pest is very difficult to control so reducing it to the peach tree might improve the odds. But ultimately I might sacrifice the tree - except I read that the bugs overwinter in soil and/or crevices in structures......what about cracks in bark, etc. too? So the battle may not be that easily won.