Cut each SweeTango® apple into 4 slices, making sure to cut off the rounded sides of the outside slices so they can lie flat. Remove all seeds. Insert a wooden popsicle stick in the bottom of each slice and carefully push it about halfway in.
In a medium microwave-safe bowl, mix together the bittersweet chocolate and 1 teaspoon of coconut oil. Zap at 50 percent power in 30-second blasts, stirring in between, until melted and smooth, 1 to 2 minutes. If the chocolate isn't thin enough to coat the slices, stir in a bit more coconut oil.
In another medium microwave-safe bowl, combine the white chocolate and 1 teaspoon of coconut oil. Melt as stated in step 3.
Spoon the white chocolate over 4 of the apple pops and coat thoroughly. Let excess chocolate ooze back into the bowl and place the pops on the baking sheet.
Using the bittersweet chocolate, repeat with the remaining 4 apple pops.
Let the apple pops sit until the chocolate begins to thicken but not harden, 5 to 7 minutes.
While the apple pops are resting, spoon the leftover bittersweet chocolate into a disposable pastry bag or small resealable plastic bag. Do the same for the white chocolate. Snip a tiny corner off each bag. If the chocolates have become too firm, a 10-second zap in the microwave should soften them.
Gently squeeze the bag of white chocolate and draw 3 concentric circles on the dark chocolate pops. Repeat on the white chocolate pops using the dark chocolate.
Drag a toothpick through the circles, from the center to the edge, to make a spider web pattern, wiping the toothpick each time. Stash the apple pops in the fridge until firm, 15 to 20 minutes.