PLA starch bags

PLA plastic is made  from plant-based materials such as corn starch. You can find about more about starch derived plastics here.
Once again they are compostable and certified as such.

Same, same but different?
PLA bags are almost but not quite the same as conventional plastic bags. However they share enough similarities to make them a very useful substitute and can be a great tool in your plastic free armoury.
They are not quite as strong as conventional plastic bags so are not good as carrier bags but they are water proof for a limited amount of time, ( eventually there is seepage), but you can certainly use them to carry fish or meat home. 
They are compostable and despite what folks might say they can be composted in a garden compost bin. Read more about PLA and composting here.

Biobags & Others
One example of PLA bags is sold by the company Biobags. Back in the early days they were amongst the first to sell  in the U.K. They have very good credentials. Their bags are certified to BS N13432 standard and are manufactured sustainably at facilities certified to ISO 9001 and ISO 14001. BioBag is the world’s largest brand of certified biodegradable and compostable bags and films made from Mater-Bi®. Biobags are the bags used in Modbury the plastic free town.

What is compostable? To be classed compostable, items must biodegrade within a certain time (around the rate at which paper biodegrades), and the resulting biomass must be free of toxins, able to sustain plant life and be used as an organic fertilizer or soil additive. For a man-made product to be sold as compostable, it has to meet certain standards. One such is the European Norm EN13432. You can find out more here.