We use a collapsible (plastic) shopping basket/tub-with-handles for wet produce, the stuff the grocery store insists on spraying periodically, and things like tomatoes where we don't care if they get wet. The store clerks are used to it now and prefer it because they don't have to scan through bags and just put the produce back in the basket afterwards.

If you go this route, keep onions and garlic separate. They last longer if they stay dry.