Correct, what you're seeing is external. I have some Novus plastic polish that I'll try after attempting to wash the remnants of the coating off, as Mayakovski suggested.So the coating is on the external side then and not the internal side?
If this is the case, I would just get a good plastic polish and polish the remaining coating off. It will certainly look better and be readable that way.
Then just keep a coat of carnuba wax on it and you should be good to go.