Fortunately, there is an easy answer. The line tamer should have a small drain hole in the middle of the base. Plug it with a cork, then add a little water -- just enough to cover the base. This will keep your fly line wet. I am not sure why this works, but it may be because the line is lubricated with water and consequently doesn't stick together. This technique works when shooting line from any container. This concept will even work from a boat deck if you strip the fly line onto a wet towel.
It is also important to remember to place more line in the tamer than you will actually cast. This helps feed line into the tamer when you begin a retrieve. If you shoot all the line, then you have to start the retrieve by placing the rear of the line inside the basket.