Combine the cream cheese, sour cream, sugar, lemon zest, half and half, vanilla and salt in the blender. Blend for a minute or two, until smooth. Add the berries and puree until smooth once again. Pour into ice cream machine and freeze according to manufacturer's instructions.
While the ice cream is churning, melt the chocolate chips and coconut oil in a small glass dish in the microwave for about 1 minute at 50% power. Stir together until smooth. Let cool. When the ice cream has finished, scoop half of it into a freezer safe container. Drizzle with the cooled chocolate sauce and swirl in with a knife. Layer the rest of the ice cream on top and repeat with the chocolate swirls. Freeze at least 2 hours, until firm. After freezing overnight, this is a very firm ice cream, due to the water content of the berries. Let it rest on the counter for at least 20 minutes prior to scooping. Enjoy!