Preheat oven to 350°. Make soft-serve: Line a rimmed baking sheet with parchment paper. In a bowl, gently toss berries with 1/4 cup sugar to coat evenly. Arrange on lined baking sheet, then bake until softened and aromatic, about 10 minutes.
Whirl berries and juices in a food processor into a smooth purée. Rub purée vigorously through a fine-mesh strainer over a small bowl and discard seeds and pulp; set purée aside.
In a medium saucepan, stir together remaining 1/2 cup sugar with the xanthan gum. Add 2/3 cup water and the corn syrup and whisk vigorously until very smooth (don't fret over a few lumps). Set pan over medium heat and cook, stirring often and adjusting heat if necessary to prevent a simmer, until sugar fully dissolves, 3 to 4 minutes. Remove from heat and whisk in berry purée and coconut cream.
Transfer soft-serve base to an airtight container and chill until well chilled, about 4 hours, or for even better texture and flavor, 24 hours.
Freeze soft-serve base in an ice cream maker until softly scoopable, about 25 minutes. Transfer to a metal bowl and put in freezer while you prepare fruit.
In a shallow bowl, combine berries and simple syrup. Let stand, stirring occasionally, until berries are juicy, about 10 minutes.
If you like, scoop soft-serve into a large pastry bag fitted with a wide star tip and twist top of bag closed. If using sugar cones, set upright in glasses and pipe soft-serve into cones in a spiral. Or spoon soft-serve into bowls. Spoon berries and syrup on top. Serve immediately.
*Find xantham gum at natural-foods stores such as Whole Foods Market.
Make ahead: The soft-serve, frozen airtight up to 2 weeks.