Cut the chicken breasts in half crosswise for quicker cooking. Set the cream cheese out to soften; cube it for easier melting.
Select Sauté on the Instant Pot. Add the diced onion and a splash of the chicken broth. Cook 2–3 minutes, stirring, until the onion softens. Stir in the garlic, cumin, oregano, chili powder, coriander, and cayenne; cook 30 seconds to bloom the spices.
Pour in the remaining broth and scrape up any browned bits on the bottom of the pot. Stir in the beans, green chiles, corn, salt, and white pepper. Nestle the halved chicken breasts into the mixture.
Lock the lid and set the valve to Sealing. Pressure cook on High for 12 minutes. Allow a natural release for 5 minutes, then quick-release any remaining pressure.
Transfer the chicken to a cutting board and shred with two forks. Meanwhile, use the back of a spoon to mash some beans against the side of the pot to thicken the chili slightly.
Ladle about 1/2 cup of hot liquid from the pot into a bowl. Whisk in the softened cream cheese until completely smooth and no lumps remain. Stir this mixture back into the pot, then add the sour cream.
Return the shredded chicken to the pot. Switch to Sauté and cook 2–3 minutes, stirring gently, just until the chili is creamy and steamy (do not boil). Stir in the lime juice.
Taste and adjust salt or heat as needed. Let the chili rest on Keep Warm for 5 minutes to thicken before serving.