I bought some screen material, some small magnets, and fender washers at a hardware store and made a screen door for the upper part of the doorway in our A128. Total cost about $15 US. Once I cut the screen to size to overlap the opening, I reinforced the edges with The Handyman's Secret Weapon (duct tape). I hot-glued 11 fender washers in places along the sides and bottom edges, then epoxied the small magnets around the door frame in such a way that the washers are pulled onto the magnets when you let the screen down. I just roll it up when we don't need it and secure it to the padded area over the door with Velcro strips.
To go through the door with the screen in place, I just walk through. The screen flows up over my head and falls back into place behind me, like those instant screen doors advertised on TV. It makes a snap as the magnets and washers find each other again, so I don't have to look to see if it closed.
Warning: this kind of door WILL muss up your hair!