Place black raspberries, granulated sugar, and water in a small saucepan. Cook covered on medium to medium high heat for 4 minutes, stirring twice. Juices will start to release.
Remove lid. Cook uncovered for 5 minutes, stirring occasionally. Take off heat. Mash raspberries with a potato masher (emersion blender would also work). Continue to cook uncovered and stirring occasionally until liquid reduces to roughly 1/3 percent. This should take 10-15 additional minutes.
While the raspberries reduce, whip the creams. Place heavy cream and vanilla extract in a medium mixing bowl. Starting on low, beat cream. Gradually increase to medium high as cream begins to thicken. Beat until cream is smooth, fluffy, and holds a stiff peak. This should only take a minute or two. Refrigerate.
Place the entire content of cream of coconut can in a large mixing bowl. Starting low and increasing to medium high, beat for 1 1/2 to 2 minutes. Refrigerate at least 30 minutes or until coconut cream thickens
Once the black raspberries have reduced, pass it through a fine mesh sieve and into a bowl (use a rubber spatula to press juice through sieve). There should be about 3/4 cups of puree in the bowl. Discard seeds remaining in the sieve. Refrigerate the puree for at least 30 minutes to thicken slightly.
To assemble ice cream, fold the chilled whipped cream into the coconut cream. Drizzle black raspberry puree into the cream mixture. Gently fold until the cream mixture is marbled. Add chocolate chips and shredded coconut. Continue to fold until cream mixture is one solid color.
Pour ice cream into a 2 quart container (I use a 10-inch loaf pan). Cover tightly with plastic wrap and a layer of foil. Freeze overnight for best results.