VT-I had this problem years ago on a bear hunt.
The tether strap was acting like a funnel, dumping water right down my neck! I had a couple of long sleeved camo t-shirts with me, so I just tied one around the tree above the tie off point. T-shirt got soaked, but my neck stayed dry.
owlbait-from your description, I am guessing you are wearing your harness under your outer layer of clothing....at least I do. I picked up a fleece scarf that is brown (there are probably some available in camo) and I just wrap it around my neck, between my neck and tie off strap and tuck it down my hunting shirt. Really warm and quiet, as my strap sometimes caught on my shirt collar.
Something to consider-I watched a treestand safety video that recommended wearing the harness on the ouside of your hunting clothing to facilitate ease of rescue. Just got me to thinkin' a bit. Good luck and I hope these suggstions help you out. Mike