It's been a painful period of learning, and several of my lanterns were damaged due to rain.
For the longest time, I was just using watered down white glue and white wrapping tissue. The first layer ripped and tore, but eventually held. It got easier from there.
As for glues. . . .
Unordered List ItemWhite glue. It's by far the cheapest, but it's not at all waterproof and gets brittle and eventually the tissue tears. I tried mixing it with driveway sealer (again, cheap), which dealt with the brittleness, but didn't help with the waterproofing.
Unordered List ItemPVA Glue - A more expensive white glue containing PVA or Polyvinyl acetates. At the very least, you should be using this type of glue rather than just cheap white glue. Also comes in the form of wood glue, which has a nice yellowish tint to it.
Typically, the first layer uses Paverpol, followed by Mod Podge or white glue. Last layer would be Mod Podge and once dry, I'll cover it all with an Acrylic Matte Medium to make it waterproof or at the very least, very water resistant.