It used to require an entitlement for an iOS app to have a transparent background, showing through to the user's wallpaper. It looks like that's not the case anymore as of iOS 13. By setting the key UIApplicationIsOpaque to NO in your Info.plist, it's possible to have an app with a transparent background.

By using a visual effect view, it's possible to have an app with a blurred, vibrant background based on the user's wallpaper. That technique is demonstrated in this app.

Disclaimer: It's confirmed that this is a bug, so Apple will probably reject your app.

Huge thanks to Tim Oliver for pointing this out.