In a small saucepan add blackberries, water, 1 tbsp sugar, cornstarch, and lemon juice.
Cook and stir until bubbling and berries have softened. Using a fork or potato masher, smash the berries.
When berry syrup can coat the back of a spoon, remove from heat and strain into a container. Store syrup in the fridge till ready to use.
ICE CREAM
In a medium saucepan add milk, cream, sugar, yolk, and salt. Cook on medium low till cream mixture bubbles.
Mix in vanilla and remove from heat. Add in cream cheese. Mix until cream cheese is mostly incorporated. Leaving a few small chunks is ok, adds a tasty texture to the ice cream.
Chill in a shallow wide container, refrigerate till cool. About 1 hour. Move to the freezer for 40 minutes. Remove and stir.
Freeze 30 minutes and stir about 5 to 6 times. This removes the large ice crystals. If you have an ice cream machine go ahead and use instead.
Stir in the blackberry syrup before the ice cream is too frozen so it can be stirred. If you want it more incorporated stir when less frozen, if you want ribbons stir in later.