You build a "coop" with hardware cloth walls instead of solid walls. Then add a part of a wall (as in a couple of feet higher and lower than the roost) on the side that storm winds usually come from. Then add a part of a wall at right angles to that back wall. Add a roof, preferably with wide overhangs (like a foot or two). That is all you need to keep the rain off the chickens.
You could do full walls instead of half walls if you leave a wide gap between the top of the wall and the underside of the roof. And/or you could do walls on both sides instead of one side. Both of these options will make the coop less comfortable for the chickens in your climate but still be much more comfortable than a typical coop or shed. It would look more conventional which can be important.
Nests could be done similarly anywhere in the coop. Or you could use one of the solid-wall parts as if it were part of a typical coop.
The "coop" could be the entire run. Or a more carefully secured part of the run. Or a hardware-cloth-screened bump out from the side of your run.