Put split peas in a medium-sized saucepan, and rinse at the tap, rubbing peas between your fingertips. Tip off water, and repeat 3 to 4 times until water runs relatively clear. Then add 4 cups of water to the pan and bring to a boil over medium-high heat. Scoop out and discard any foam that rises to the surface.
Drain potatoes and add them, along with turmeric, to split peas. Give a stir or two, then reduce heat to medium-low and cover the pan. Cook, stirring occasionally, for 20 to 25 minutes; potatoes should be cooked, and peas should be tender but still intact.
Meanwhile, heat a small skillet over medium-high. Once skillet is hot, add chiles, coriander and cumin. Toast for a minute or two, shaking skillet every few seconds until chiles blacken and seeds are fragrant and turn reddish-brown. Transfer to a blender along with the tomato. Purée, scraping down blender, as needed, to make a smooth paste.
When pea/potato mixture is done, add some of the pea cooking liquid to the blender and purée again for a few seconds. (This helps loosen the paste, assuring you'll get every last bit.) Pour contents of blender into saucepan. Stir in cilantro and salt.
Increase heat to medium-high and boil vigorously for 12 to 15 minutes, uncovered, stirring occasionally to let flavors blend, and thicken.
For a thicker mixture, mash some of the peas and potatoes with the back of a spoon. Serve warm over rice.