Squeeze the soaked tamarind in its soaking water to extract all the juices. Strain the smooth, sour liquid into your cooking pot, discarding the pulp and seeds.
Add the chopped tomatoes, turmeric, asafoetida, and salt to the tamarind liquid. Mash the tomatoes with your hands to blend them into the water. Simmer on medium heat for 8-10 minutes until the raw smell of tamarind is gone.
While simmering, coarsely crush the peppercorns, cumin seeds, and unpeeled garlic in a mortar and pestle.
Stir the crushed spice paste and 2 cups of water into the simmering pot. Bring the heat down to low.
Watch the pot carefully. As soon as a light froth covers the entire surface of the liquid, turn off the heat immediately. Do not let it boil.
In a separate small pan, heat ghee or coconut oil. Add mustard seeds. When they splutter, add the dried red chilies and curry leaves. Pour this hot mixture into the rasam, cover with a lid immediately, and garnish with coriander before serving.