I was confused because what your describing sounds like large parts of New Zealand and Tasmania, so I checked Wikipedia which says:

Temperate rainforests occur in oceanic moist regions around the world: the Pacific temperate rainforests of North American Pacific Northwest as well as the Appalachian temperate rainforest in the Appalachian region of the United States; the Valdivian temperate rainforests of southwestern South America; the rainforests of New Zealand and southeastern Australia; northwest Europe (small pockets in Great Britain and larger areas in Ireland, southern Norway, northern Iberia and Brittany); southern Japan; the Black Sea–Caspian Sea region from the southeasternmost coastal zone of the Bulgarian coast, through Turkey, to Georgia, and northern Iran.