It is possible, I have heard that some of the acoustically transparent screen around are actually made from windows shades or some such material. It believe they are fabrics with a very open weave so that sound can pass through.
That being said, I wouldn't really recommend it. Regular DIY screens will give pretty decent performance if you have a flat smooth surface and a good matte screen paint. However, an acoustically transparent screen is another animal. The degree of acoustical transparency is going to be difficult to determine, if it was just making all frequencies softer that wouldn't be so bad. The problem is that the screen will attenuate higher frequencies first, so you'll start to lose your treble.
You could evaluate the fabric by draping it over your speakers to see how it changes the sound, however I think you'll have a hard time finding a good balance between light loss through the perfs and good sound. Also, I don't think you will be able to find anything with a gain characteristic *and* acoustic transparency.