In a medium bowl, whisk whole milk, evaporated milk and condensed milk to combine and set aside.
In a medium saucepan, bring rice, cinnamon sticks and 1 ½ cups of water to boil over medium heat. Reduce the heat to a simmer and continue to cook, uncovered, until the water has almost completely evaporated, about 10 minutes.
Stir in the cream mixture, sugar and vanilla and bring back to a boil.
Reduce the heat to a simmer and add lime zest. Stir every 2-3 minutes, to keep the mixture from scorching and sticking to the pan, for 40-50 minutes, until thick but still creamy.
Remove from the heat and discard the cinnamon sticks and lime zest. Let the pudding rest for 10 minutes.
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