I've had the same questions. For the fans I have looked at, they claim to be satisfactory for a wet area but nothing said about being water tight. My conclusion is that they are suited for outdoors, under a roof, but not exposed to direct rainfall.

So - - - I have not installed a fan yet. If I proceed I believe I will obtain a large piece of plate glass to go on top of the pergola as a roof over the fan.

I'll be interested in any additional / differing information you obtain.

Good luck.

I have thirteen years experience with "outdoor ceiling fans", in this case made by Craftmade. Theyre supposedly designed so a foggy wet environment wouldn't damage them, but of the four I purchased one was replaced after 6 years, now another one is iffy. The main bearing apparently gets some corrosion making it run a little rough, and the switch gets pretty rough to operate. So my overall impression is they don't live up to the marketed impression.

I believe that most outdoor fans are rated for damp locations, not wet locations. There's a difference an a message in the electrical wiring section would give you a definitive answer but I think a good way of thinking about it is that you'd need to protect it from direct contact with water. In the case of a pergola, a solid roof that doesn't leak.

Of course, I could be all wet. Make sure you put it on a GFCI protected circuit.

Ceiling fans are given different ratings. A damp rated fan can be used outdoors if it is sheltered under a porch. A wet rated fan should be used if your outdoor fan is not well sheltered - as with a pergola. You have to read carefully because sometimes information tends to be vague, such as "Acceptable for outdoor use." Umm, what does "outdoor" mean here?