Original User: ernosson
Peter,
I think it's not a coordinate system problem.
Your data is more accurate than the topographic basemap provided by Esri. If you compare your data to the Imagery basemap, it can be noticed, that the shapes of the islands are different, but the outline of the main island in the northwest is lined up with the basemap.
So the "problem" is that your data was digitized based on another raster dataset and the basemaps are bit different.
Also try compare the data to Bing Maps. You can add it to the Desktop by clicking File >> Add Data... >> Add Data from ArcGIS Online >> Search for Bing Maps.
I attached the transformed shps.
I hope this can help.
Regards,
Akos