Pencil tile.

Generally makes for a sloppy look and it is very difficult to find a way to cap off the ends when they are that thick. There are some wide schluter strips but that would draw attention to the strips and not the backsplash. A backsplash is an area where permabase is not necessarily required so I would repair any damage to the sheetrock and mount the tile directly on the sheetrock. Use appropriate adhesive for this application. If you feel that you really need permabase, then I would remove the sheetrock and install the permabase in its place and then mount the tile. A small pencil liner, a bullnose or a schluter strip could then be used.

I've attached a pic so you can see what I'm working with right now.

Beautiful tile. You can adhere the tile directly to the sheetrock without adding additional backerboard (this can be done on a backsplash but should not be done in a bathroom, on a floor or in any wet area). You just need to use the correct adhesive for that. Generally would use a pencil liner to finish the ends as shown above, but you will have to cut the tiles in order to get a straight line. Good luck on that - an arabesque is a devil to cut. You may be able to find a schluter strip in the right color to cap off the end and it would be less noticeable. You would need to visit a tile store to see the sizes and color options of schluter.