Pat the chicken dry with paper towels.
Heat the oil in a large pot over medium heat.
Add the chicken and brown both sides. Work in batches if the pot feels crowded.
Add the onion, ginger, and garlic. Cook for 2 to 3 minutes until fragrant.
Add the fish sauce and stir well.
Pour in the chicken broth and water.
Bring the soup to a boil, then reduce the heat to a gentle simmer.
Cover and cook until the chicken is nearly tender, about 30 to 40 minutes depending on the size of the pieces.
Add the chayote and continue cooking until it is fork-tender, about 10 minutes.
Add the leafy greens and cook for 2 to 3 minutes, or until wilted.
Taste the broth. Add more fish sauce if needed.
Ladle the hot Tinola into bowls and serve with steamed rice and calamansi.