Preheat oven to 325 degrees F.

Prepare an 8-inch square baking pan by lining the pan with aluminum foil, leaving 3-4 inches of foil draping over the sides of the pan.

The Crust:
Place graham crackers and gingersnaps into the bowl of a food processor and pulse until the crackers form fine crumbs, about 30 seconds. To the crumbs, add butter, brown sugar, flour, and salt. Pulse until the ingredients are fully combined. Using the bottom of a measuring cup or your fingers, press the crumbs into the bottom of the prepared pan.

Bake until the crust is beginning to brown, 12 to 15 minutes. Let the crust cool on a cooling rack for about 30 minutes.

The filling:
In a large bowl, beat cream cheese until it is very smooth, about 3 minutes. Scrape the beaters and sides of the bowl frequently. Gradually beat in granulated sugar until the mixture is combined, about 1 minute. Beat in eggs, one a time, making sure that each is fully incorporated. Beat in sour cream and vanilla extract until fully combined, about 30 seconds. Scrape the cream cheese mixture into the baking pan, over top of the cooled crust. Spread evenly.

In a medium saucepan, combine 1/2 cup granulated sugar and 1/2 cup water. Simmer, stirring frequently, until the sugar dissolves. Add the chopped rhubarb and the piece of peeled ginger to the water and sugar syrup. Simmer, stirring occasionally, until the rhubarb is very soft and falling apart, 7 to 10 minutes. Remove and discard the piece of ginger. Cool for about 10 minutes.

Using a teaspoon, spoon 1/2 cup dollops of the rhubarb mixture gently onto random places over the surface of the cheesecake. Using a small knife, gently swirl the rhubarb mixture into the cream cheese mixture.

Bake until the edges are set but the middle still jiggles slightly when the pan is shaken slightly, 35 to 40 minutes, rotating the pan halfway through baking.

Place the pan on a cooling rack and allow the bars to cool for 2 hours. Cover with aluminum foil and refrigerate for at least 3 hours, and up to 24 hours.

Using the overhanging foil, lift the bars from the pan. Gently remove the foil and cut into 16 bars.

Top the bars with the remaining rhubarb sauce immediately before serving.