I'm non-dominant. I was unfortunately taught to shoot everything right handed, had a pro instructor try to convince me I was left eye dominant when I was in high school and shooting with a league... But I was non-dominant all along. After instructing myself now for over 15yrs, I'd much rather be "cross dominant" than non-dominant, as I'd simply shoot the same side as my dominant eye and be happy with life. Unfortunately, my non-dominant condition means whichever direction my eyes are pointing, relative to my centerline, is the eye which takes over. So when I cheek into a long gun sight, or take any offset step with a handgun, I get the WRONG eye in control.
It's not so complicated for me. I played with the tape, it works to some extent, and I've used the (true meaning) co-witness trick since I was a kid, but really all it takes is a slight furrowing of my offside eyebrow to kick the control over to my sighting eye.
I've found it handy once I got the hang of it - in high school, before the "pro" instructor tried to convince me it was impossible to be non-dominant. Anything I can shoot right handed, I can shoot as well left handed - any difference between the two is solely a matter of currency of practice with either hand.
As an instructor, I fervently encourage shooters to learn to shoot with their dominant eye side. Eye dominance is a hard-wired neurological pathway, whereas shooting technique familiarity and automaticity ("muscle memory") is a developed skill. There's no "handedness" for shooting, as there's not enough microscale motor control involved to be overly dependent upon our dexterous handedness. I can appreciate there are a lot of those "I've been doing it for 50yrs," or "so-and-so pro shooter was cross dominant," or "do what feels most comfortable for you," but there's also a dude with no arms with archery records shot with his feet, and Jimi Hendrix played a right handed guitar upside down whenever he jammed with Albert King - neither are evidence enough for me to start shooting my bows with my toes, or flip my guitars upside down.