Special equipment: a 15-by-18 sheet pan
For the crust: Position a rack in the middle of the oven and preheat to 350 degrees F.
Mix the cracker crumbs with the butter, sugar and salt in a medium bowl until evenly moistened. Press the crumb mixture into the bottom of a 15-by-18-inch sheet pan. Use the back of a measuring cup to smooth out the crumbs. Bake the crust until golden brown, rotating the pan halfway through, 10 to 12 minutes. Cool the pan on a rack, 10 minutes.
For the filling: Lower the oven to 325 degrees F. Blend the cream cheese and sugar in a food processor until smooth. Add the eggs and 1 cup sour cream and pulse until well combined. Pour the batter into the prepared pan and smooth the top with an offset spatula.
Bake until the cheesecake is slightly puffed on the edges (the edges will deflate as the cheesecake cools) and just set, rotating the pan halfway through, 40 to 45 minutes. Completely cool the cake in the pan on a rack. Cover with plastic wrap and refrigerate 8 hours or up to overnight.
Combine the raspberry jam with 1 to 2 teaspoons water in a small bowl until spreadable but not thin. Spread the jam on the left third of the cheesecake. Top with the raspberries.
Spread the remaining 1/4 cup sour cream over the middle third of the cheesecake and top with the mango slices.
Spread the blueberry jam over the right third of the cheesecake and top with the blueberries. Slice into bars for serving.
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