Ah yes, Apache tears, lovely little drops of obsidian. These are about the most difficult of all materials to tumble polish! I remember my first batch, it took me about 4 months of experimentation to get it right. The following is how I mastered tumble polishing Apache Tears:
First, be sure that you use LOTS of plastic pellets throughout every run, from coarse through fine, as you do not want a single conchoidal chip.
The secret is in the polish stage. Carefully place the tears in the polish barrel and before you add the water, saturate the water with white table sugar (and I mean saturate!!). This forms a really thick syrup that when added to the tears and pellets will further the "cushion" needed to protect these little glass pieces. Note: Be sure to add enough pellets for the barrel to be 4/5 full!
Then add cerium oxide, about half again the amount you would normally use, a dash of Ivory flakes, and run for about 10 hours before checking. Enjoy!
Answer contributed by Dale "Cougar" Armstrong