I'd never heard of these before, but in Europe they apparently use "fly curtains" as an alternative to screen doors. These are essentially just beaded curtains, or occasionally patterned strips of plastic, with the strands placed very close together; flies perceive it as a surface rather than an aperture.
The advantage of these over a screen door is simply that you don't have to open and close it, and pets can come and go.
Different patterns, colors and opacities of bead are offered for both aesthetic and light-admitting purposes.
You can also order them with prints.
To remove them for cleaning, the rail they're suspended from can be easily popped off of its mount. Alternatively, they can be cleaned in place. Soapy water will do the job.
The examples you see here are from Dutch manufacturer Liso.