From what I've gathered, things can be a bit more difficult if you have a more exotic identity document and either want to use one of the cheaper MVNO providers

From what I've gathered, things can be a bit more difficult if you have a more exotic identity document and either want to use one of the cheaper MVNO providers (which generally don't operate any physical stores at all), or aren't in a position to visit the physical branch of one of the major carriers (Telekom, O2, Vodafone). In that case customer identification is commonly done via the post office's Postident system (either at a physical post office, or online via video) or via some alternative video chat system, and those might not work for all passports from all countries worldwide.