Rinse the rice in a sieve under cold running water.
Place rice in a bowl and add enoough cold water to cover the rice at least 1 inch.
Let rice stand overnight or at least 8 hours. (Don't skip this step!).
In a heavy saucepan, combine the rice and the 4 cups water. Cover pan and bring to a boil. Lower heat and maintain a gentle simmer until all the liquid is absorbed and rice is tender but not mushy. This takes 1 1/2 to 1 3/4 hours. The texture of the pudding should be very thick but still spoonable. Toward the end of cooking, be sure to stir occasionally to prevent any scorching or sticking. If the rice looks as though it is too dry, add a little more water.
Remove from the pot and stir in the sugar and salt. Let cool slightly. Then add the lime juice.
For Coconut sauce:.
In medium saucepan,combine the coconut milk and salt. Simmer, stirring until the liquid has thickened slightly, 15-20 minutes.
Pour into a small bowl and let cool.
To serve, spoon rice pudding into individual bowls. Spoon coconut sauce over each serving and let each person stir it into the rice as they eat. Can be eaten cold but warm is best.