Preheat oven to 350°. Toast almonds on a small rimmed baking sheet, stirring occasionally, until golden brown, about 4 minutes. Let cool.
Using a sharp knife, remove peel and pith from blood oranges, Cara Cara orange, navel orange, and tangerines. Working over a bowl, cut between the membranes to release segments into bowl; reserve juice.
Puree half the mango, the lime juice, and 1 cup reserved citrus juice in a blender until smooth.
Gently toss citrus segments with remaining diced mango.
Top mango puree with cut-up fruit. Sprinkle with cacao nibs and toasted almonds.