I am working with this natural earth shapefile. The dateline disappears after using the mentioned clip to hemisphere plugin. Densifying the geometry does not help.
Weird to me seems that clipping to a hemisphere centered at lat -76,98267 / lon -163,36119 works just fine (dateline present after clip) while after clipping to a center at lat 26,50588 / lon -171,82873 the dateline dissaperes although it should be visible.
Solved as noticed in the comments by @Jake a bug in the clip to hemisphere plugin is the problem. I quickfixed it and its all good now.
180in https://github.com/jdugge/ClipToHemisphere/blob/master/ClipToHemisphere.py#L264 to180.1in your local copy of the plugin (in.qgis2\python\plugins\ClipToHemispherein your home directory), then restart QGIS. I'll fix the error and upload a new version of the plugin on the weekend. – Jake Dec 01 '16 at 16:45