Warm the olive oil in a large skillet over medium heat.
Sauté the diced onion until softened, about 4 minutes.
Stir in the minced garlic and cook until fragrant, 30 seconds.
Season the chicken with taco seasoning, salt, and pepper.
Nestle chicken into the skillet and sear 2 minutes per side.
Add green chiles and chicken broth. Cover and simmer 12–15 minutes.
Transfer chicken to a board and shred with two forks.
Lower heat and whisk cream cheese into the skillet sauce until smooth.
Return shredded chicken. Fold in sour cream and lime juice.
Simmer 2 minutes to thicken. Adjust salt and pepper.
Warm tortillas until pliable in a dry skillet or microwave.
Fill tortillas with creamy chicken. Top with cheese and cilantro.