I'm not sure what the process is myself as my supervisor did it for me, but as a PhD student (from Canada) I'm registered at the University of Bern but I live in Zurich (and got my residence permit for Zurich), as I spend most of my time in the lab at the ETH Zurich. I commute once a week or as needed to and from Bern. I don't think it caused too many issues in terms of permits, but perhaps they need to confirm some sort of collaboration, and I'm not sure what paperwork is needed for that.