Superfish Menu links disappear on iOS (only if they have dropdown menus)
Enough tinkering and I found a solution. I'm not sure WHY it works, but it does. 

During my experiments, I wondered if the z-index was the problem and if it was getting covered by something, so I cranked the z-index up to 10000 but that still didn't solve it.

I eventually discovered the position: relative thing I mentioned above. So I opened up the Computed Style box and toggled the position: relative rule on and off, on and off, which would break it, fix it, break it, fix it. Then I looked in the Computed values to see if anything else changed. Sure enough, the z-index changed. When it was in the working state the computed z-index was auto, so I created a css rule to explicitly set z-index to auto for those elements and viola... it's working.

Hopefully this will be helpful to someone else.