Rather than focus on alternatives to plastic lids on disposable containers, I thought it would be beneficial to have another solution entirely:
Use reusable lunch gear
As a consumer, you can bring a bowl or soup cup with you (or even just a tupperware container with a lid), and ask that your takeaway soup be placed in that instead.
KeepCup offers a ...
Plastic films can line paper containers.
But also plastic is being developed that will bio-degrade in any compost situation.