Best places outside the U.S. to honeymoon for experiencing culture, great food, and avoiding mass tourism.

Our budget is about $6000. We originally thought about Italy, and haven't ruled it out, but the crowded tourism is a turn off and subtract from the cultural experience. I know most honeymoon locations will have some levels of this, but we want to keep it on the low to medium at level at most. Also not wanting to visit a location that doesn't want tourism.

She does have a family member who has property in Spain (not Barcelona or Mallorca) where we could stay, but id love to come up with more ideas to propose. She has traveled far, far more than I have, like France, Mexico, Columbia, so id like to propose something new. Japan would be a dream, (we'd want to see rural areas, cherry blossoms, hot springs for people with tattoos, as well as modern things), but im told it is pretty expensive.

We hope to be able to stay somewhere where we can not be too crowded, and can easily access many different things to do or locations to visit (both nature and man-made), but good food, leisure, and culture is a must.

Thanks for your ideas!