Peel the bananas and cut each one into two or three segments, about two inches long. Pat every segment completely dry with a paper towel, then insert a popsicle stick about halfway into one flat end of each piece.
Lay the banana segments on a parchment-lined baking sheet and freeze for 30 to 45 minutes, until they are completely hard with no give when gently squeezed.
Combine the dark chocolate chips with half the coconut oil in a microwave-safe bowl. Microwave in 30-second bursts, stirring between each, until smooth and glossy. Stop before the chocolate gets hot enough to scald.
Take the bananas out of the freezer in batches of three or four at a time. Dip each segment into the chocolate, turn it with a fork to coat all sides, and let the excess drip back into the bowl.
While the chocolate is still wet, press two candy eyes onto the front of each mummy. Work immediately; if the chocolate sets first, the eyes will fall off.
Return the sheet to the freezer for 10 minutes until the dark chocolate is firm.
Melt the white chocolate chips with the remaining coconut oil in 20-second bursts, since white chocolate scorches faster than dark. Pour into a zip-top bag and snip a tiny corner off the bottom.
Let the white chocolate rest until warm rather than hot, then drizzle it back and forth across each mummy in a few diagonal passes, leaving a gap around the eyes. Uneven bandages are exactly right for a mummy.
Freeze the finished mummies for 15 minutes until the white chocolate is completely set. Keep them frozen right up until serving.