Add the frozen berries, sugar, and lemon juice to a small saucepan. Cook over medium heat, stirring occasionally, until the berries break down and the mixture thickens into a jam-like consistency, about 12-15 minutes. It should be thick enough that a spatula dragged through the center leaves a trail for a second before the jam runs back together.

Transfer the compote to a heatproof bowl and refrigerate until completely cooled.

In a medium bowl, whisk the white cake mix and milk together until smooth and no dry streaks remain. The mixture will seem thin at first, but as it rests, the cake mix will absorb the liquid and thicken to the consistency of cake batter. Refrigerate while you prepare the ice cream base.

In a large mixing bowl, beat the softened cream cheese until smooth and creamy. Add the sweetened condensed milk, vanilla extract, almond extract, and salt, then beat again until fully combined and no lumps of cream cheese remain. Scrape down the sides of the bowl as needed.

In a separate large bowl, beat the heavy cream with an electric mixer on medium-high speed until stiff peaks form. The whipped cream should hold its shape when you lift the beaters without drooping.

Add about one-third of the whipped cream to the cream cheese mixture and gently fold it in. Fold in the remaining whipped cream in two additions, using a rubber spatula until no streaks remain. Be careful not to overmix, or you'll lose the air that gives no-churn ice cream its light, creamy texture.

Spoon about half of the ice cream base into two 32-ounce deli containers or a 9×5-inch loaf pan. Dollop half of the cooled berry compote over the top, followed by half of the cake batter. Repeat with the remaining ice cream base, berry compote, and cake batter. Use a butter knife or skewer to gently swirl everything together a few times, being careful not to overmix.

Cover tightly and freeze for at least 8 hours or overnight until firm. For the creamiest texture, let the ice cream sit at room temperature for 5-10 minutes before scooping.