In a large bowl, combine the goat meat with vinegar and salt. Let it marinate in the refrigerator for 1 hour to remove the gamey taste. After marinating, rinse the meat thoroughly with clean water and drain well.
Fill a large pot with enough water to cover the meat and bring to a boil over medium-high heat. Add the marinated goat meat and boil for 7-10 minutes until you see scum rising to the surface. Drain the meat in a colander and rinse both the meat and pot with clean water.
Put the clean pot back on the stove over medium heat. Return the cleaned meat and add 6 cups of fresh water. Bring to a boil while skimming off any scum that rises. Once the broth is clear, add the quartered onion, crushed ginger, and fish sauce.
Lower the heat, cover the pot, and simmer for about 1½ to 2 hours, or until the meat becomes tender but still firm. Remove and discard the ginger pieces.
While the meat is cooking, prepare the tamarind. Wash the unripe tamarind and place in a separate pot with 2 cups of water. Bring to a boil and cook until the tamarind becomes soft and the skins begin to burst. Mash the tamarind with a fork.
Strain the tamarind mixture through a fine-mesh strainer set over a bowl. Keep mashing with a fork, adding some cooking liquid back to get all the juice out. Throw away the seeds and skins, then pour the tamarind juice into the pot with the meat.
Add the finger chilies and cook for 4-5 minutes. Taste the soup and add salt as needed. Add the chopped green onions and cook for just 1 minute more.
Serve your soup steaming hot in bowls with rice on the side. If you like, serve with a small dish of fish sauce and calamansi for extra flavor.