You are "processing" the data in FME, QGIS and ArcGIS - but how are you viewing the data? For example, the FME Data Inspector will show the data correctly, but only until you turn on a background map, when it will appear as above. So what application is the above screenshot from?
Also, the format is going to be important I think. For example, if you have Google KML datasets, then FME would write data correctly, eg:
<coordinates>175.0,-10.0,0.0 175.0,10.0,0.0 -175.0,10.0,0.0 -175.0,-10.0,0.0 175.0,-10.0,0.0</coordinates>
...but Google Earth won't show the data correctly. However, if you wrote the data as:
<coordinates>-185.0,-10.0,0.0 -185.0,10.0,0.0 -175.0,10.0,0.0 -175.0,-10.0,0.0 -185.0,-10.0,0.0</coordinates>
...then Google Earth is more than happy to show that feature!
So I think it's a combination of coordinate system (as @MappaGnosis says), how well the data format supports this (and how are you creating it), plus whether the application you are using is actually capable of interpreting and rendering this properly at all.