Such a critter has existed?

I mean, a two-speed hub without a coaster brake and the ability to back pedal?

That would interest me, hugely.

I remember my favorite bike of all time, before I discovered fixed gear bikes.
It had a Bendix two-speed and a coaster brake.

If I could have a two-speed hub fixed gear bike, I can't imagine it getting any better than that.

Except for a single-speed fixed gear bike.