Pour the water, tangerine juice, and lemon juice into a saucepan and place over medium-high heat. Bring to a boil, add the honey and sugar, and stir until dissolved. Pour the mixture into a shallow metal pan and let cool to room temperature. Place in the freezer and stir every 30 minutes for 2 to 3 hours. You'll want the granita to be firm but not rock hard.
Place the cherries in the bowl of a food processor and process until smooth. Put all the ingredients into a saucepan and place over high heat. Let it come to a boil, reduce the heat, and simmer for 10 minutes. Pour into a heatproof pitcher and let cool to room temperature.
Divide the granita among 4 martini glasses, top with fresh raspberries, and drizzle the whole thing with the syrup.
Recipe courtesy of Dan Smith and Steve McDonagh