Fresh lime zest and juice are best for this tart. If you can't get Key limes (sometimes called West Indian or Mexican limes), use regular (Persian) limes, or even lemons!
Preheat oven to 350 degrees. Finely grind crackers with sugar in a food processor (you should have about 1 1/4 cups). Add butter and pulse until combined. Transfer mixture to a 9-inch tart pan with a removable bottom; press into bottom and up sides. Place pan on a baking sheet and bake until crust is fragrant and slightly darker, 10 to 12 minutes. Transfer to a wire rack; let cool completely.
Meanwhile, puree mangoes in food processor and strain through a coarse sieve. In a bowl, whisk together 1 1/4 cups strained mango puree, condensed milk, lime zest and juice, yolks, and salt. Pour into crust. Bake until filling is set around edges but still slightly loose in center, about 25 minutes.
Let tart cool on a wire rack 1 hour; then refrigerate until chilled, at least 2 hours and up to overnight. Serve, with whipped cream and diced mango on the side.