In the days of aluminum pans, I found that boiling some citic acid got rid of the dark discoloration. But I'm afraid to try it with my tinned copper lest it do something weird (like the baking soda I tried on my Caphalon).
BTW: The Barkeepers does a fair job in getting rid of the worst part of the copper crud. Not bright and shiny like a good polishing, but makes polishing a lot easier if you care to do it. I don't try to keep a high shine and even like a moderate copper patina, but I don't like it to get really dark.