Empty the cranberries into a saucepan and transfer ½ cup to a small bowl.
Add 1 and 1/3 cup erythritol, 1 strip of lemon zest, and 2 tablespoons of water to the pan and cook over low heat, stirring occasionally, until the erythritol dissolves and the cranberries are soft, about 10 minutes.
Increase the heat to medium and cook until the cranberries burst, about 12 minutes.
Reduce heat to low and stir in the reserved cranberries.
Add more erythritol and salt to taste and cool to room temperature before serving.