Combine all the ice cream base ingredients in a blender. Blend until smooth. Pour into an ice cream machine. Churn according to the manufacturer’s instructions.
Meanwhile, make the cream cheese icing swirl. Beat the softened cream cheese in a medium bowl with an electric mixer until fluffy. Beat in the cream until incorporated. Add the glycerin and sweetener. Mix until light and fluffy.
When the ice cream is done churning fill the container you will freeze it in about halfway with ice cream. Put a blob of icing on top. Fill until the ¾ mark with more ice cream. Add another blob of icing. Use a butter knife to swirl the icing in gently. I just ran the knife back and forth a few times. Finish filling the container with ice cream. Freeze until firm. (Repeat until you've used up all ice cream and icing).
This makes a generous 1.5 quarts or 3 pints. I filled three pint containers and we had enough left to taste test.