We've spent a lot of time looking into alternatives to plastic bags for those leafy greens that wouldn't get to the consumer in an edible state without them. We've trialled various bags over the years but we've come to the same conclusion as others that have had more time to do the research than us, namely that alternatives that are better for the planet aren't yet available (this is wrong but it's a result of how joined-up thinking between production of bags and disposal of bags does not exist!).
Here's a couple of quick points below, but please read the links below that to get the in-depth info
Paper bags are not the answer, as producing and shipping often uses more energy than producing and shipping plastic bags. Of course, they are good for the end-user as they can be recycled or composted easily. Also, they don't offer the same benefits that plastic bags do for leafy greens.
Most compostable bags are certified as such if they are compostable under industrial composting conditions and there are not currently any industrial compostable conditions in the UK. And most people's home compost heaps will not be good enough to compost such bags. And, of course, you have to take into consideration the materials used to make the compostable bags, and whether land was used to grow bags that would be better used to grow food.
I hope this doesn't sound like a cop-out - we will continue to reduce our plastic use and look into alternatives. If you read all the links, you'll see that Riverford are trying to get home-compostable bags by 2020 - we'll keep an eye on what they manage to obtain (they've got more resources to look into it than us).
https://www.infinityfoodswholesale.coop/packaging_faq/plastic_statement/
https://www.growingcommunities.org/blog/2018/11/packaging-update-reuse-reuse-reuse
https://www.riverford.co.uk/aboutus/thoughtful-packaging/home-compostable-plastic