Cut the honeydew melon in half and remove the seeds. Press the seed pith through a fine mesh sieve over a bowl, and reserve the resulting liquid. Discard the seeds. Cut the melon into sections, and remove the rind.
Place the melon and reserved melon liquid into a tall pot. Puree with a stick blender. Alternately, you can puree the ingredients in a blender. Add the coconut milk, then continue blending until smooth. Stir in the melon liqueur.
Remove the small tips on each sprig of mint, and reserve for garnish. Stack the remaining mint, then begin rolling the mint into a ball, securing with kitchen string. Drop the rolled mint into the pot with the melon soup. Cover and refrigerate for 2 hours or overnight.
Remove mint before dividing the soup into chilled cups. Garnish with the reserved mint tips.
