I can only guess you want to do this because it'll look cool - which i agree - a projected image against a semi transparent surface would be cool.
Unfortunately, I don't think its gonna work.
Problem 1: pico projector. These things aren't ment for long term high quality use. Quick, in a pinch, a salesman goes to an office and wants to show a bigger image without relying on bringing a full blown projector or the client having one. Coffee shop or restaurant presentations also come to mind. Project against the wall at the table from a laptop.
The restriction is lumens. Projectors rely on a back light (some sort of halogen in older ones, led in newer ones). The greater the lumens, the brighter the picture.
Low lumens would be "ok" in a dark room, but you'll need the lumens to be much higher if your going to leave the lights on or the room has windows.
Couple this with the fact that you want the screen to be semi-transparent.
Projection screens (proper ones) have silver in them (hense, Silver Screen back in the day). You can get away with white, but the silver in the screen helps reflect the light.
You "might" be able to do this if you used a real projector (a proper sony or epson) but their bigger than a pico projector. Again, make sure it has a high lumen.
Problem 2: For the screen, translucent white for sure. But if you can see through it (like i think you want) you might not have enough light hitting the surface of whatever (glass, plexi) to produce the image. LCD and LED screens work in that their tiny lights (rgb) packed (crammed) together to produce an image. Ever seen a dead pixel? one of those lights got stuck or died or whatever. Projecting against a translucent screen, it'll be like the dead pixels are everywhere.
You turned down the idea of using a film. Maybe white window tint (see a car tinting place) on a piece of glass or plexi.
O, O, O!!! Some higher end cars (and even some mid range) have HUD's. A projector that sits behind the steering wheel and projects speed, fuel, etc... on to the windshield. That is accomplished with a special film on the glass and a higher end projector.
Or go crazy (and expensive) and maybe try to have some glass made like windows i've seen where when an electrical current is applied to the gas in the glass, it'll turn white (opaque) and you cant see through it. Frame it nicely and when its off, its just a box with glass in it.
or, if its really just for looks, theres a way to take a photo of whats behind a monitor and use it as the wallpaper on the desktop and make the screen "appear" transparent, but this also relies on the proper viewing angle.