Aluminium (or better yet, copper) tape is a good way of shorting out any charge build-up on the perspex case. It is also a great way to create a short circuit and fry all your electronics inside the case.
I have seen plastic cases with ESD-sensitive electronics inside, protected by a thin layer of electronics protection type conducting foam on the inside. This is generally only on the underside of the board, which is likely to come into contact with the plastic case. This foam is the type that ICs are sometimes shipped in, dark gray or black and very... foamy. Resistance is high, of the order of several Megaohms per centimeter, but that works fine.
Another more home-grown alternative, is to cut up and use a metallized shielding plastic bag, the type with a wee bit of conductivity courtesy a thin layer of conductive material sandwiched between layers of plastic. These bags are often semi-transparent, so you'd retain the ability to peek into the case, as the perspex probably lets you do. Again, some vendors ship electronics in such bags.
Using either of these two approaches will not create a Faraday cage, nor will they entirely eliminate static charge build-up. What they will do is distribute such charge across the foam / plastic, thus spreading the risk, so to speak, and reducing the chances of a very high charge build-up.
Note that ESD protection bags look similar, but without the metallic shine, and are not equivalent to metallized plastic. Those will not be useful in this context.
For more information on the metallized bags and ESD bags, see this answer to another question: "Do antistatic bags have conductive interior, exterior or both?".