1 tablespoon lime juice cordial, such as Rose's Lime Juice
Pour the grenadine into a saucer. Invert a stemmed wineglass into the grenadine to coat the rim of the glass. hold the glass sideways by its stem over the sink. Sprinkle the outside edge of the glass with sugar, twirling the glass to coat just the mostened outside rim with sugar. Set aside to dry a pit. (If you plan to serve a bunch of these, the glasses can be prepared ahead.) Combine the remaining grenadine and the rest of the other ingredients in a blender and whir on high speed until the mixture is thoroughly pureed. Turn off the blender.
With the blender running, slowly add more ice cubes, no more than three at a time. Listen carefully to the sound of the machine. When the machine resumes its normal pitch rather than sounding gravelly, look at the drink. You will notice a vortex in the center of the mixture. Keep adding ice until the vortex disappears and the mixture looks smooth.
Turn off the blender and stir the mixture with a long spoon or a chopstick. Replace the lid and turn on the blender. The vortex will reappear. Keep adding ice cubes, two or three at a time, until it disappears.
Once again, turn off the blender, stir, replace the lid and blend for about 10 seconds, or until the sound of the motor tells you that all the ice has been thoroughly chopped. Pour into the prepared glass. Serve with 2 long straws.