Since it is fog and not rain that you are really concerned about I would put the tarp on the hay and then pull it back on sunny days. You want to try and avoid your damp hay getting warm under the tarp, that is when it will mold.
Where do you live in No. CA? Are you in the central valley? I seldom see fog in Anderson, I am on a mountain and sometimes there is fog along I-5 but next to never at my home. I LOVE IT. I lived in Fairfield (west of Sacramento) for 23 years and hated that valley fog. I worked in Sac. so sometimes commuting was a real challenge. Actually stayed home a few days rather than risk my life.
I know where that is...Yep, you get fog...sometimes for days on end.
The most important thing is air circulation. If my hay is damp when I stack it it will mold on the bottom but the sides and top will be fine. If the hay is the least bit damp be careful that the tarp is off when it gets warm or it it is in direct sunlight. Also, look at the hay when you feed it and make sure that there is no mold.
another weird idea that might work, if you do close the room up some to try to avoid the air getting in and getting everything moist, maybe try getting a big salt block and storing it with the hay (not touching, just in the same area), salt somehow sucks moisture out of the air... when it's humid at my barn, the salt blocks are just soaked to the point of water running down them... I would assume that that will also help get the moisture out of the air...
We have never worried about fog here , we do get thick fog from the great lakes here.
I think the easier solution is to salt your hay...we salt ours. Just toss a hand full of rock salt or road salt under the hay bale and then a hand full on top of the bale. The salt will fall out as the animals eat but will keep dampness off the bales.
We also salt bales that end up a little to wet from baling before it is dried down enough like the last field we cut it kept raining just light sprinkles so the hay wouldnt dry out enough so we baled it, stacked it with 6 inch spaces between each bale in a single layer then salted it and we turn the hay every 3 days and check for any hay heating up. So far out of 200 bales 8 have heated, as soon as it starts warming we feed it out before it can mold or cause problems...the livestock loves this wet hay though it is like fresh cut grass.