I can't imagine what it could hurt to keep a picture of your passport on your phone (but not only in the cloud, because you may not have reception where you need it).
We have never been asked for passports except for border control or when we know ahead of time they'll be needed (getting driving licenses, registering vehicles, etc.). Not once. We are, officially, required to show our residency documents upon demand but have never unexpectedly been asked to show those, either.
We leave our passports and residency documents in our safe the entire time we're on the island. We've never bothered to take them when we're on excursions to the mainland, either.