I did some glass tests. See below.
The first image shows the known problem. After I read Matt's explanation I edited the glass geometry and deleted the inner wall of the upper part of the glass. I tried some glass shader experiments with this edited glass from picture 2 on.
Picture 2 has some weird refraction, so I checked "doublesided" in the glass shader from picture 3 to 5. In picture 6 I unchecked it again and reduced the IOR that was 1.3 in pictures 1 and 2 to 1.2.
A bit confusing, I know, and now I'm a bit uncertain, which one looks best.
Of course the glass in the first image looks best, except the black stuff...