Berry Purée: Combine blackberries, confectioners' sugar, and water in a saucepan over low heat. Cook, stirring frequently, until it comes to a boil and thickens slightly, about 2 minutes. Transfer to a bowl, mash gently with the back of a spoon, cover, and refrigerate until ready to use.
Ice Cream Base: Blend the cream cheese, milk, granulated sugar, vanilla, salt, and corn syrup until smooth. Add the heavy cream and whisk to combine. Cover and refrigerate until cold, at least 2 hours or up to overnight.
Churn: Whisk the base to recombine, then freeze and churn in an ice cream maker per manufacturer's instructions. When it's finished churning, swirl in the chilled blackberry purée. Serve soft right away, or transfer to a container and freeze 2 to 3 hours for a firmer scoop.
