Heat oil in a skillet over medium-high heat; add onion and jalapenos. Cook, stirring frequently, until tender, about 5 minutes. Add garlic; cook for 1 minute more and remove from heat.

Transfer jalapeno mixture to a blender. Add water, diced tomatoes, hot green chiles, mild green chiles, and cilantro. Cover and blend on high until mostly smooth.

Pour blended mixture into a large pot over medium heat. Add processed cheese, chicken soup, cream cheese, hot sauce, cumin, and chili powder. Cook and stir until completely melted, 5 to 7 minutes. Stir in lime juice; reduce heat to low.