A package for preserving respiring produce contained in the package wherein the package defines a package volume for containing a portion of the produce and a package atmosphere, which comprises: - a packaging material which comprises a Biaxially Oriented Poly Ethylene (BOPE) or a Mono Directed Oriented poly Ethylene (MDOPE)-containing polymer film with a haze of at most 10, preferable at most 5, more preferably at most 3,and most preferably at most 2, as determined by ASTM D 1003, - said polymer film being provided with one or more perforations enabling gas exchange with the atmosphere surrounding the package to form the package into a Controlled Atmosphere Package; wherein: - the package has a package carbon dioxide transmission rate (CO 2 TR pack) - wherein the one or more perforations provide a perforation carbon dioxide transmission rate (CO 2 TR perf) such that the package carbon dioxide transmission rate (CO 2 TR pack) is the sum of the perforation carbon dioxide transmission rate (CO 2 TR perf) and the material carbon dioxide transmission rate (CO 2 TR mat) of the packaging material (CO 2 TR pack = CO 2 TR perf + CO 2 TR mat) and wherein - the CO 2 TR pack is at least 1000 ml/24 hrs per 100 gram produce to be packed, preferably at least 1500 ml/24 hrs, more preferably at least 2000 ml/24 hrs, most preferred at least 2500 ml/24 hrs. - a packaging material which comprises a Biaxially Oriented Poly Ethylene (BOPE) or a Mono Directed Oriented poly Ethylene (MDOPE)-containing polymer film with a haze of at most 10, preferable at most 5, more preferably at most 3,and most preferably at most 2, as determined by ASTM D 1003, - said polymer film being provided with one or more perforations enabling gas exchange with the atmosphere surrounding the package to form the package into a Controlled Atmosphere Package; wherein: - the package has a package carbon dioxide transmission rate (CO 2 TR pack) - wherein the one or more perforations provide a perforation carbon dioxide transmission rate (CO 2 TR perf) such that the package carbon dioxide transmission rate (CO 2 TR pack) is the sum of the perforation carbon dioxide transmission rate (CO 2 TR perf) and the material carbon dioxide transmission rate (CO 2 TR mat) of the packaging material (CO 2 TR pack = CO 2 TR perf + CO 2 TR mat) and wherein - the CO 2 TR pack is at least 1000 ml/24 hrs per 100 gram produce to be packed, preferably at least 1500 ml/24 hrs, more preferably at least 2000 ml/24 hrs, most preferred at least 2500 ml/24 hrs. - the package has a package carbon dioxide transmission rate (CO 2 TR pack) - wherein the one or more perforations provide a perforation carbon dioxide transmission rate (CO 2 TR perf) such that the package carbon dioxide transmission rate (CO 2 TR pack) is the sum of the perforation carbon dioxide transmission rate (CO 2 TR perf) and the material carbon dioxide transmission rate (CO 2 TR mat) of the packaging material (CO 2 TR pack = CO 2 TR perf + CO 2 TR mat) and wherein - the CO 2 TR pack is at least 1000 ml/24 hrs per 100 gram produce to be packed, preferably at least 1500 ml/24 hrs, more preferably at least 2000 ml/24 hrs, most preferred at least 2500 ml/24 hrs.