Place the water, galangal, shallots, lime leaves, and lemongrass into a wok, and boil for a few minutes to release the fragrances.
Add all of the vegetables, and continue to boil until al dente.
Stir in the coconut milk (if using), and season to taste with the salt, sugar, and lime juice. There should be a balance between sour, salty, and sweet. If not using coconut milk, just continue with the seasoning.
Add the nam prik pao and coriander. Taste, and adjust the seasonings if necessary.
Continue to boil the tom yam for another minute.
Serve immediately, garnished with a few slices of spring onion and some torn coriander leaves.
Although I've given quantities for the seasonings (salt, sugar, lime juice, and nam prik pao), you may find you need to adjust them to suit your own taste.
If you don't like a lot of heat, start out with one teaspoon of nam prik pao, taste, then add more if desired. Conversely, if you like it really spicy, add more nam prik pao.
If you want to be really Thai, serve tom yam with a separate bowl of steamed rice. I really like to drizzle the soup over my rice, while some of my friends add the rice to the soup. Some folk don't do either!
If you want to eat the aromatics, they won't do you any harm but it's not usual to consume them!
Nutritional information does not include the optional coconut milk.