Overpass API and overpass turbo really are not the right tools for this job, as you cannot query yet for any non-closed areas in the first place (it's not implemented yet). This makes the whole endeavour very challenging.
What you see in overpass turbo originates from the internal OSM -> GeoJSON conversion, but it's not possible to take that logic into account for any kind of query.
I suggest to take a look at keep right! instead. It has a "non-closed area" detection. Osmosis Osmose QA probably has something similar.
Example: [Keep right! example][1] -> will mark the following [way][2] as erroneous.
[1]: http://keepright.at/report_map.php?zoom=17&lat=48.15152&lon=16.29759&layers=B0T&ch=0%2C30&show_ign=1&show_tmpign=1
[2]: http://www.openstreetmap.org/way/349493111