Combine mangoes, cream, brown sugar, and corn syrup in a blender or food processor. Blend on high for 30 seconds.
Transfer to an airtight container. Stir in spiced rum and salt. Refrigerate, 8 hours to overnight.
Pour mixture into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.