Don't have Real Atlantic here, so can't comment on them.

Sobey's, Safeway Sunterra and Co-op supermarkets (Calgary) all provide small, light plastic bags for the fruits and vegetables; fresh behind the counter meats all get wrapped by hand and are blood-free. Their unfrozen meats in the cold racks are all well-sealed and all blood-free. Rarely shop at Superstore (a little cheaper but too few checkouts and no help taking purchases to the car) but haven't seen leak problems there.
Some large plastic-sealed meats (e. g. local leg of lamb) show traces of blood in the package, but no leaks.

These stores all have the larger 'green' bags in which purchases are packed, so only the large plastic bags are charged for. Never have to wash them (except when they accumulate too much dog hair in the car).

gnu, change stores!