What's the best way to remove a rusty, broken screw from a tiled bathroom wall?
You could try a screw extractor, i have had moderate success with them.  

You use the first end clockwise to make divot in the screw so that the second end that turns counter clockwise will have something to bite its teeth into. Go slow and apply pressure into the screw. In your case the head is missing so it makes it very difficult but worth a try. 



The set does not cost to much and is good to have on hand anyways even it does not always work.