"not a fan of shades" doesn't leave much. heavy curtains that you can pull back the rest of the time, or tinted glass film of some sort. you might give top-down accordian blinds at least one glance.

Ukrainian Village Master Bath · More Info

I like the idea of shutters, but they don't open like regular louvered shutters, or are there any like that?

Why not just drapes

That's another option, but would have to be a clean look.

Wood blinds are a great look and give you privacy and light control. Another option is natural woven shades made of bamboo, grass, or matchstick, lined with plain white fabric.

I used a film, which is applied with soapy water, and it took care of our problem.