VoIP App for iPhone (universal, for standard SIP protocol)
I know that there is an app that behaves similar to your mention. The case is, that the app I'm talking about is well known in DACH and is only working with routers of a specific company 'avm'. The app is called 'Fritz!App Fon' and empowers you to receive and make calls. Even when not opened, the app receives incoming calls. Unfortunately you need those routers 'Fritz!Box' to take advantage of that feature. VOIP providers can then be set in the router's settings and be bound to your device - very simple and not complicated at all.

Edit: Via VPN you are able to make those calls from any place on the world - internet connectivity required :)