Here are a couple ideas to chew on. I can't give a specific answer.
1. Autocomplete has an open event which can contain a function. Its is described in the documentation. Conceivably the function could look for the specific iOS keyboard event and open/instantly re-open the dropdown. I don't know if the dropdown would retain its item contents, if the user removes the mobile keyboard after the dropdown is populated.
2. The issue is mostly about Javascript's "bubble up" behavior for events. Lots of info is on the internet about how to deal with that behavior. Here's a starting point that lots of people have found helpful.
http://stackoverflow.com/questions/1357118/event-preventdefault-vs-return-false