Use powder...

• tablets contain anywhere from 2x to 4x the detergent you need
• tablets only work in the main wash cycle and so you have no detergent for the first wash cycle
• powder is substantially less expensive

Are you dead set on tablets? If not you can just get the powder. It’s the same thing as what’s in the tablet.

I order mine from https://smol.com/uk. They come in a cardboard packet which fits through the letterbox. I was also fed up with the ridiculous and unnecessary amount of plastic in the supermarket equivalents.

Checkout fill refill, who do powder in a big container. When it's empty, you send the big tin back to them and they refill it send it back to you.

https://www.fillrefill.co/product/dishwash-powder-5kg-home-refill/

Aldi still have the cardboard box and cellular film tablets. As with Lidl, these do run out quite regularly. I've started getting a box whenever I see them, rather than when I run low, just in case. I've only rarely had to resort to the pouches.

I just get the powder from Sainsbury’s.
I find I get better results, as I can kinda dial in how much is needed. It means I can put some in the pre wash too.
Just remember that if you're using powder, the salt/rinse aid need filling separately.

Morrisons own brand tabs only have the cellulose wrapper, if that's what your definitely after.

If you haven't already, try and find a zero waste shop near you, the kind of place where you take your own containers and fill it up with what you need.

Me and my partner made it a part of our routine to visit once a month, and we have successfully replaced a bunch of our household items with zero plastic options.

Aldis ones are much the same as Lidl, cardboard box and the tablets are just inside

Do you have powdered detergent in the UK? Try that. I switched over entirely. You just need to experiment with dosage a little.

I just use powder, it’s way cheaper and just as effective, also you can be more economical with it as you don’t need a full tablets worth of powder for most washes. The only thing I miss is the powder I have is designed for commercial use and is unscented - doesn’t affect the cleaning but the chemical smell isn’t the best when you open the dishwasher. But on the other hand you’re not paying for perfume you don’t actually need - also you avoid the environmental impact in perfume manufacturer too.

(edit, for clarity, I'm in the US)

I just use Nellie's dish powder. They're Canadian, but it just comes in a big 1.5L tin. The only plastic is the reusable scoop, and the local refillery will refill it when you run out. Check local refilleries in your area, they are a bit more expensive, but generally have high quality products and little to no plastic waste.

Have you heard of dishwasher sheets? There's a few companies making them now, these are the Australian brand I use sheets

I use Finish brand, individually wrapped in rice paper - looks like plastic but dissolves in the water. Many others are the same. I doubt that plastic wrapped ones are trying to trick you, they tell you to remove the plastic - they don't want to get complaints about clogged dishwashers. In fact this is probably the reason most manufacturers use the rice paper, people dodn't read instructions. You can also get a box of powder you pour into the little cup.