It's the same as polishing/sharpening, etc - going "down the grits". You might try this, but beware that shininess doesn't mean good optical performance, so you might find, for example, that it lets you down in strong sunlight.
1. Clean thoroughly with soft materials. I've used my similar mask to shield my face and glasses when jet-washing and when spraying water based paint. This comes off remarkably well with bio washing powder (contains enzymes) Hat-tip to Rafezetter for that one! Put a fairly strong solution on, then clean off with a clean paintbrush and lots of flowing, warm but not boiling hot water. Make absolutely sure you lift any grit or metal particles as soon in the process as possible, as they will add scratches later.
2. Coarse polish with T-cut (for car bodywork), applied with a J-cloth or similar. Make absolutely sure all cloths used are clean, as the shield's surface is such soft plastic. Wash off with baby shampoo or "sensitive skin" bubble bath/shower soap, as neither has many additives. Washing up liquid contains glycerine which is a really bad idea as it leaves a sticky residue. It's worth trying a small area, to see if it's working: do it; wash off and dry; then take a critical look.
3. Fine polish with Dura-Glit wadding (rebranded Brasso recently). Similar caveats - if it's dried up in the tin don't use it. You can use a little water lubrication with this too. You can also use liquid Brasso but it's a bit coarser and prone to drying out in scratchy lumps.
3a. Jeweller's rouge is the next step down, used as a thin-ish paste in water.
4. Finally (LAST RESORT) I would be tempted to try this:
https://www.everbuild.co.uk/product/pvc ... t-cleaner/
BUT it is a seriously nasty chemical (you need barrier gloves - e.g. PPE ones, which it will probably perish), and it works by slightly melting the surface, so might actually destroy the transparency - TEST a small area in a top corner of the shield. Use some old well-washed cotton sheet as a cloth to do this.
Bear in mind it's all about working the surfaces down around scratches. You're not actually "removing" scratches at all. It's a big task with a big area, and you are in essence thinning and weakening it. Only you can decide when it's good enough for practical use.
I also had success with car headlights last year using water-lubricated fine wet+dry followed by T-cut and Brasso/Dura-Glit (beware - uPVC cleaner clouds them!), but it's tougher than those face shields, thicker, and rigid material, and you aren't looking through it.
When all is said and done, how much is your time worth? Doing this right takes a while -
you may decide buying three or four more ready-cut-out replacement polycarbonate sheets is much more cost-effective. But if you want to go ahead, that's how I would approach it, personally.
E.