I think "from mainland" or something similar is correct instead of mentioning "culturally Han". The current culture difference was mainly formed over the process of having different rulers for both places in the 20th century. "Han" is a way broad notion for us Chinese. A fair amount of Chinese derived from "Han", an ethnic group.

The reason I want to point this out is "Han" means a lot more than the current status quo, which was transformed badly after the ruling party dominated the main land.