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 similar idea for coffee/tea cups. I have one, and use it daily for tea/coffee from my regular cafe. That cafe also supplies keep cups to their customers, for a small fee - the customer can either return the cup when they're done and get their fee back, or keep it for their own use.
The founder of KeepCup has actually started another business focused on the issue of takeaway containers:
Aptly called Returnr, the business lets customers pick up their food in a reusable stainless steel bowl after paying a $6 dollar deposit. After eating, users can then return the bowl, re-use the bowl, or keep it forever (and forgo their $6).
But founder Jamie Forsyth tells StartupSmart he’s finding most users are opting for the second choice, re-using the bowl numerous times in an effort to avoid relying on single-use containers.
So in a few years, it might be possible to simply request a reusable container from your local eatery over a disposable one.
Note - I have no affiliation with KeepCup and they're far from the only supplier - I'm just using them as a relevant example.