In a saucepan heat all ingredients over low heat. Stir frequently until it boils, about 5 minutes. Reduce for another 2 minutes to let everything meld together and then remove from the heat. Transfer to a shallow container, mash berries with a fork or spoon and refrigerate.
For the cheesecake ice cream base:
In a bowl, beat cream cheese, heavy cream, sour cream and sugar just until smooth. Add zest, vanilla and salt. Combine, transfer to a container with a lid, or bowl with plastic wrap, and refrigerate until very cold, 6 to 8 hours or overnight.
Place the mixture in your ice cream maker and churn according to manufacturer´s directions.
Transfer to a freezer-proof container, add strawberry mixture, swirl a few times, cover and freeze until solid.
Keep in mind that homemade ice cream freezes quite solid, so you'll want to take it out to soften a bit before serving.