I'm sort of ambidextrous and yet solid left eye dominant - ain't no way around it.
Archery - lefty
Shotgun - lefty
Rifle - lefty
Handgun - lefty, but was trained/re-trained to be ambi, both eyes open. *
* I must be a freak of nature.... If I hold my 1911 .45ACP lefty, my left eye takes over. If I switch positions and my stance, and shoot righty, my right eye takes over.
With archery, I have to shoot lefty, lefty eye open, right eye closed. If I try shooting both eyes open, I see two targets or I see two sets of sight pins. Same with my shotgun. I see two targets or two beads/barrels. With my rifle, it doesn't matter as I have a Bushnell 3-9x40 scope and can shoot that both eyes open, lefty, with my left eye lined up in the scope tube.