trentOH, The back of the strap actually is dyed, just lightly. It is a neatsfoot oil blend with Fiebings Pro oil dye added. Just enough to feed the leather and give it color. It is then sealed, once thoroughly dried. I have rubbed heavily with a cloth and nothing comes off.
In my the posts on my last bag, the question came up and I addressed it. Suffice it to say, I have no idea how folks dye the back of leather dark brown, have it look good, and not have it come off on clothes. I am a novice, still learning, so hopefully that secret will come to me one day.
Best wishes, and God Bless, Marc