In a large pot, warm the oil over medium heat. Add the white/light green parts of the scallions, carrots, garlic, and ½ teaspoon salt. Cook until softened, about 3 minutes, stirring occasionally.
Add the water, coconut milk, potatoes, mushrooms, ginger, soy sauce, fish sauce, and remaining ½ teaspoon salt. Stir together, then nestle the salmon in, skin-side-up. Press down any salmon or potatoes that are sticking out so that they're covered with the liquid. You can add up to another ½ cup water, if needed.
Bring to a boil over high heat, then reduce the heat to medium-low to maintain a simmer for 5 minutes or until the salmon is cooked through (easily flakes with a fork in thickest part, depends on thickness). If salmon is done, remove and let cool. If not, simmer for another 1-3 minutes and check again.
Once the salmon is done, continue to simmer for another 5-10 minutes or until the potatoes or rice are cooked through.
Turn off the heat and stir in the greens and lime juice.
Remove the skin from the salmon and flake it up with a fork.
Divide the soup into your bowls and top with the flaked salmon, dark green scallions (add cilantro if you have it), and a squeeze of lime.