I remember that in Germany you can buy bags made from corn starch, but they're very thin and very fragile. You need to be really lucky to bring home your shopping bags safely and you definitely cannot reuse them. But you can put them on the compost, which is really cool! If they work on making it a little more robust, bags like that would be my first choice
Producing a paper bag has environmental impacts equivalent to three lightweight polyethythene bags. For cotton bags, it's about a 130x multiple. The best option is to reuse a thick nonwoven plastic bag until it's worn out; woven plastic or cotton bags are rarely re-used often enough to repay their greater costs of manufacture.