The day before you plan to churn, freeze the bowl of your ice cream maker.
Peel and slice mango. In a food processor or blender, puree mango until smooth. Measure out 1 cup of puree and transfer to a large bowl. Add sugar, water, tequila, lime juice, and a pinch of salt and whisk until sugar dissolves. Chill for one hour in the refrigerator, or 20 minutes in the freezer, stirring every 5 minutes to prevent the mixture from freezing.
If making spicy strawberry jam instead of chamoy, whisk together jam and Tajin in a small bowl. Taste and add more Tajin if desired.
Saturate a paper towel with tequila, then rub it on the inside of your ice cream maker’s freezer bowl (this prevents the sorbet from freezing on the sides of the bowl).
Freeze mango mixture according to your ice cream maker’s instructions until the texture of soft serve, likely around 20 minutes. Transfer spoonfuls of mango sorbet to a resealable container, alternating scoops of sorbet with tablespoonfuls of chamoy or jam mixture. When everything has been transferred to your container, use a butter knife to swirl two mixtures together.
Cover and freeze until solid, about 4 hours. Serve topped with more chamoy and/or Tajin if desired.