In a large pot over medium heat, warm the oil. Add the onion, cilantro stems, jalapeno, and ½ teaspoon salt. Cook until the onions have softened, about 3-5 minutes, stirring occasionally.
Add the cumin and cook for about 30 seconds, stirring often, to toast the cumin. Stir in the water, chicken, bell pepper, cabbage, tortilla chips, and remaining 1 teaspoon salt. Bring to a boil over high heat, then give it a stir and reduce the heat to medium-low. Simmer rapidly for 15 minutes (uncovered), stirring halfway through. Turn off the heat and transfer the chicken to a cutting board. Shred up the chicken and return it to the pot.
In a high-speed blender, add the remaining cilantro, lime, garlic, and ½ cup cold water and blend on high until smooth. Immediately pour back into the soup and stir to combine. Taste and season with more salt or lime, if needed.
Serve topped with plenty of cheese, crumbled tortilla chips, yogurt or sour cream, and chopped tomatoes (if using).