How do I set the Apple ID for the App Store?

I have two iPhones, one personal and one business. Early in the setup process, I made the mistake of signing in with my personal Apple ID instead of my work Apple ID.

I changed the Apple ID to the work one on the Apple ID screen and set it for Media & Purchases, but when I try to update my (work-specific) apps like Slack EMM or Boxer, the App Store is asking for the authentication info for my personal Apple ID. How do I set the "App Store Apple ID" to the "iPhone Apple ID"?