We installed the simplest of blinds which drops down from the outside of the ceiling / roof to the outside of our NE facing top, 3rd level balcony about 8 years ago made out of some sort of synthetic fabric that has lasted unbelievably well and changed our lives, really. It virtually gave us another, cooler room in summer. (Note: blind drops vertically, no metal bars.)
We had the set-up professionally measured and installed, took the consultant's advice and it was all a very worthwhile experience. Happy to give you the brand: Supershades (edit it out JT if you need to).
Our unit is not visible from the street, just neighbours' freestanding houses. We did it without permission. Naughty us.