Start by preparing the spice paste. In a blender or mortar and pestle, grind the shallots, garlic, coriander seeds, and candlenuts into a fine, aromatic paste.
Transfer the blended spice paste into a large cooking pot. Add the bruised galangal, salam leaves, water, and coconut milk to the pot. Stir the mixture well to ensure that the coconut milk is fully incorporated and bring it to a gentle boil over medium-high heat.
Once the broth begins to boil and release its fragrant aroma, add the corn pieces, cubed tempeh, and melinjo beans if using. Stir gently to combine and allow the ingredients to simmer.
Season the broth with salt and sugar, adjusting to taste. Continue cooking the mixture for about 10 minutes, allowing the flavors to infuse.
After the initial simmering, add the chayote, eggplant chunks, and long beans into the pot. Stir well and let the vegetables cook for another 5 minutes, just until they are tender yet still vibrant.
Finally, stir in the melinjo leaves along with the sliced red and green chilies. Give the soup one last gentle stir, then immediately turn off the heat to preserve the color and freshness of the greens.
Serve Sayur Lodeh warm, allowing the creamy, coconut-infused broth to complement the medley of tender vegetables and savory tempeh.