In Earth Not a Globe a semi-transparent moon is suggested based on astronomical observations. While a semi-transparent moon is no longer supported by the current incarnation of the Flat Earth Society, it is important to keep compiled evidence of such phenomena for posterity.
In the transparent moon accounts various stars have been seen to occult (pass in front of) the moon. This is completely impossible in the Round Earth model. Such observations remain a mystery to this very day. It has been suggested that the moon is semi-transparent. However, an alternate explanation for these observations might be that the moon is not semi-transparent and some (but not all) of the stars are simply below the altitude of the moon, therefore able to pass in front of it. This is possible in the Flat Earth models where the sun, moon, and stars are all at similar altitudes.
See the chapter on Moon Transparency in Earth Not a Globe: http://www.sacred-texts.com/earth/za/za63.htm