I have added a curtain to a pergola in the past.

The rod was attached much like a curtain rod to the inside beam of the pergola. The curtain fabric had to have vents. This is essential to stop the wind punching through the fabric. I had one that had fine square vent panels. At the base it was weighted with a thin heavy chain sewn into the bottom seam.

I have bought this triangle shaped canvas sunscreen with rings on the ends at an outdoor/camping store. You can hang it with carabine-hooks or rope, super easy. See if you can find one where you live.

Look into the option of a Shade Sail for your area. They are used at playgrounds in my sunny, hot city.

We have curtains attached to the inside with sprayed conduit with conduit mounting brackets. I use curtain tie backs to drape to the post and screw with a large washer the inside bottom corner to the post. No one sees it, but it stays put

Thank you JW. I like the washer idea. Did you use mesh material as ASV recommended or sunbrella material?
ASV- I think I know the shade cloth material you are referring to, but haven't been able to find it. (I picture it being black and easy to look through?)
I've seen the outdoor sail-types...they are very cool, but I am concerned about the wind from the west blowing it inward (I would be hanging it vertically. The problem is when the sun gets low in the sky. It then becomes unbearable, esp this last summer when it seemed every day was a record setter. (I'm in SE Wisconsin.)

Hi there, our fabric was white/ cream and solid for the most part with the square panels of mesh in it. I will look and post

Hi there
Not able to find exactly what i wanted to show you but this gives you a good idea:

this is nice for a pavilion... Is it made of glass? or better an open air?

Glad to be off assistance (from so far away) :-)

Have you thought about adding some roman shade panels in the spaces between the slats. Look at Tenshon LLC's profile on Houzz or their website www.tenshon.com. It may give you some ideas. I added one photo below.

Nina, love the look, but the problem sun time of day is when the sun is lower in the sky and shade is needed from the direct west. Blinds, shades, curtains, etc need to hang vertically.
The deck is raised (there is a walk out basement underneath). I have quite a bit of weight already on the deck with furniture, plants, etc (I keep adding more and more :-). The look of the stretched sun shade appeals to me, but if the wind blows from the west, as it usually does, combined with the weight already on the deck, I may be putting too much strain on the structural integrity of the deck.
A curtain is probably the answer, but one with vent panels for the air to pass through.
I appreciate the comments I've been getting. This is the year I'll be putting something up for sure. The photo I posted is from early last summer. Since then, I had the pergola stained to match the deck and this winter I painted the deck furniture a brownish gray to match the house and the tables are an off-black. I think this will make the colors of the flowers and pillows stand out and will give me more options for colors of whatever I use for sun protection.