Do ahead: Freeze the bowl of your ice cream maker according to the manufacturer's directions., To make the syrup: Use a vegetable peeler to remove the zest (green peel) from 3 limes in large strips; reserve these. Juice the limes to get 1/2 cup., Put the lime juice and sugar in a small saucepan and bring to a simmer, stirring until the sugar dissolves. Remove from the heat and set aside to cool., To make the base: Combine the yogurt and buttermilk; set aside. In a small bowl, combine 2 tablespoons of the milk with the cornstarch to make a smooth slurry., In a 3-quart bowl, whisk the cream cheese, half the sugar, and salt until smooth., Fill another 4-quart or larger bowl half full with ice and water., In a 4-quart saucepan, combine the remaining milk and sugar with the cream, corn syrup, and reserved lime zest and cook over medium-high heat. Bring to a rolling boil, then reduce to a simmer and cook for 4 minutes. Remove from the heat, and gradually whisk in the cornstarch mixture., Return to the heat and bring back to a boil, stirring constantly until slightly thickened, about 1 minute. Remove from the heat and gradually whisk the hot mixture into the cream cheese mixture until smooth., Add the yogurt/buttermilk mixture, lime syrup, and cardamom and whisk until smooth., Set the bowl of base into the ice water bath and stir frequently, adding more ice to the water as needed, until the base is cold, about 30 minutes., Strain the base into the frozen bowl of your ice cream maker to remove the lime zest. Freeze according to the manufacturer's directions, until thick and creamy. Pack the frozen yogurt into a storage container., Press a sheet of parchment or wax paper against the surface and seal with an airtight lid. Freeze in the coldest part of your freezer until firm, at least 4 hours., Store in the freezer for up to two months.