Stir together graham crumbs and butter until all crumbs are evenly coated. Press into the bottom of a 9-inch springform pan. (Line the inside of the pan with a piece of parchment or acetate to help the finished cheesecake come out of the pan easier.)
To make the Strawberry Raspberry Compote:
Puree strawberries and raspberries together in a blender. Press through a mesh sieve and discard seeds and solids. Pour puree into a medium-sized saucepan. Whisk together water and cornstarch and add to pot. Cook over medium heat, stirring frequently, for 15-20 minutes. Cook until jam-like in consistency and there’s about 1 cup of mixture. Let cool completely before using.
To make the Blueberry Blackberry Compote:
Puree blueberry and blackberry together in a blender. Press through a mesh sieve and discard seeds and solids. Pour puree into a medium-sized saucepan. Whisk together water and cornstarch and add to pot. Cook over medium heat, stirring frequently, for 15-20 minutes. Cook until jam-like in consistency and there’s about 1 cup of mixture. Let cool completely before using.
To make the Cheesecake Filling:
In a large bowl, cream together cream cheese and Greek yogurt with an electric mixer until smooth and there are no lumps. Add vanilla extract and powdered sugar. Mix on medium speed until all ingredients are incorporated.
In the bowl of a stand mixer (or another bowl) fitted with the whisk attachment, whip heavy cream until stiff peaks form. Fold half of whipped cream into the cream cheese mixture. Repeat with second half of whipped cream.
Divide the cheesecake filling into three bowls. Stir the Strawberry Raspberry Compote into one bowl, the Blueberry Blackberry Compote into the second bowl, and leave the third bowl plain.
The Strawberry Raspberry filling will be a pale pink and the Blueberry Blackberry filling will be a purplish color. Stir a dab of gel food coloring into each (red to the Strawberry Raspberry and blue to the Blueberry Blackberry) until the desired color is reached. If you don’t want to fool with food coloring, you can skip this step!
Spoon the Strawberry Raspberry filling over the graham crust in the springform pan; smooth with a rubber spatula. (I also tilted the pan around and shook it very gently to help the filling flatten out.) Place the other two fillings in the refrigerator. Freeze the springform pan for about 30 minutes (or until firm enough that you can touch it without it leaving much of an indentation).
Repeat with the plain filling layer and freeze. And finally, repeat with the Blueberry Blackberry filling and freeze until completely firm, 3-4 hours or overnight.
When ready to serve, let cheesecake thaw in the refrigerator for 15-20 minutes, or thaw at room temperature about 10-15 minutes. Use a warm knife to cut slices and serve with whipped cream and additional berries if desired.