"Please note that I'm not asking for scientific evidence, and that I think it's clear that lots of things exist that can't be described in a scientific way."

I don't know. I've recently come round to thinking that it's valid to ask for scientific evidence. If we allow believers off the hook of needing to provide scientific evidence we're sort of letting them put the argument entirely on their terms. I've never seen a single phenomenon that science couldn't either already explain or which science, plausibly, will one day not be able to explain within the known laws of physics (in other words, I've never seen magic). I don't see why we shouldn't hold believers to the same stricture (but I suppose I'm a hard man...)