I used sandpaper of different grits to sand the surface of plastic. The surface is smooth but it looks dull after sanding. I read that one could use toothpaste to remove the foggy look but it did not work. I also tried vinegar with baking soda. When the plastic was inside the vinegar+baking soda bath, it looked transparent and shinny. However, when I took it out to wash it with water to clean the smell, it looked shinny and transparent when being washed but it became foggy and cloudy again after washing. I put it back to vinegar+baking soda bath. It became transparent again but when I used a clean microfiber cloth to clean it, it became foggy again. Looks like some kind of thin film on it. Putting it in vinegar+backing soda bath for about 8 hours did not help. What caused the foggy/cloudy look? Would you please tell how to I make the plastic fully transparent?
The tiny scratches on the plastic surface reflect and diffract light in different directions, so edges of objects behind the plastic become distorted, and the clear areas of the plastic are less clear.
There are several ways to make the plastic more transparent.
Use a finer grit sandpaper - really really fine: 1500 grit or finer, if you can get it. Plastic abrasive sponges are more gentle and can be used wet, so may create smaller, less deep scratches.
Use toothpaste to "sand" the plastic in the first place. You have to get the paste to be pasty thick, not watery thin. It will reduce scratches, but if you don't put deep scratches on in the first place, it will be easier to shine up a plastic surface. Toothpaste will reduce the shininess of a perfect surface, but usually will improve an ordinary surface. There are commercial pastes for cleaning and shining stovetops that are more abrasive, and you can make your own with sodium bicarbonate (no vinegar) - but you have to give it some energy, some elbow grease, and it takes a while. You could use a buffer wheel on a power drill.
Filling in the tiny scratches with a liquid helps even out the surface, but water evaporates and the effect does not last. An oil (e.g., silicone) or petroleum jelly (very thin layer) might last long enough for you - days, maybe weeks.
Depending on the plastic film, you might be able to heat it slightly to allow the scratches to deform slightly and become slightly wavy instead of sharp valleys and projections. This will require a lot of experimentation to get enough heating without distorting the plastic beyond your requirements. Hot air or hot smooth metal, or perhaps hot water or steam might work.
The least likely method to work is using a solvent to smooth out the surface. You might get some immediate effect that looks like an improvement, but any solvent that dissolves the very top surface will also diffuse into the plastic and when it evaporates, you will have a plastic that is disturbed even deeper, and will be foggier. Working this approach further suggests that you could use what would be better described as a plasticizer, a high molecular weight liquid that is compatible with the plastic, but does not migrate (out, usually, but in here), buff the surface and quickly clean the plastic to remove all the solvent/plasticizer. Now, since you probably don't have access to different kinds of plasticizers, you could try some vegetable oil plus a little of a more powerful solvent, like a few percent of alcohol, or mineral spirits, or even acetone, and you could add some sodium bicarbonate for a little abrasion, and use a buffer wheel.
Use a headlight restoration kit or if you feel confident that you can use a clear gloss spray paint well this will also restore plastic clarity. You've done all of the prep that would normally be taken to restore a headlight or plastic pieces on a car but you need the final coat.