Induction coil interference from an electromagnet?
Both windings are just coils of wire -- giving them different names does not change how they function.

Yes, when wound on the same core, they will definitely interact with each other quite strongly. However, you need to decide whether that interaction works in your particular application.

I take it you want to put a DC current through one of the coils in order to provide a field similar to what a permanent magnet produces. That's fine -- the other coil will not pick up DC. However, any AC (noise) in that DC current will be coupled to the other coil, and it's possible that it will overwhelm the signal from the guitar strings that you're looking for.

So, how good are you at building low-noise DC current sources?