Make the strawberry sauce: Combine the strawberries, water, sugar and cornstarch in a medium saucepan. Bring to a simmer over medium heat, stirring occasionally. Continue to cook, stirring occasionally and gently smashing the berries as they soften, until thick and jammy, about 25 minutes. Pour into a small bowl and let cool completely, stirring occasionally. (You will have about 11/3 cups sauce.)
Meanwhile, make the chocolate sauce: Combine the chopped chocolate and coconut oil in a small microwave-safe bowl and microwave in 30-second intervals, stirring, until melted and smooth, 1 1/2 to 2 minutes. Let cool slightly.
Make the ice cream: Blend the heavy cream, sugar, cream cheese, sour cream, vanilla and salt in a food processor, scraping down the sides of the bowl occasionally, until very thick and fluffy, about 2 minutes.
Spoon about one-third of the ice cream mixture into a freezer-safe 2-quart baking dish, then spoon about one-third of the strawberry sauce over the top, then one-third of the chocolate sauce (it does not have to evenly cover). Repeat the layering two more times, ending with the chocolate sauce. Freeze until the chocolate sets, 5 to 10 minutes, then cover with plastic wrap and freeze until firm and scoopable, at least 6 hours or overnight. Let soften about 30 minutes at room temperature before scooping.