Spread apples into a 9x9-inch baking dish. Mix brown sugar, flour, rolled oats, butter, cinnamon, and allspice together in a bowl until crumbly; sprinkle topping over apples.
Bake in the preheated oven until the crumble topping is browned and the apples are soft, 30 to 40 minutes.
Combine cranberries, white sugar, and water in a saucepan, stirring to dissolve sugar. Bring to a boil, reduce heat to low, and simmer, stirring often, until cranberries pop, about 5 minutes. Remove from heat; compote will thicken as it cools. Serve apple crisp topped with warm cranberry compote.
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