It's a mix of two words: Bromance (A made-up word for men who spend quality time together) & marriage.
Bromage sounds exotic and almost French in nature.
It is for men who share a "bromance" that have been friends for awhile (a year or longer) and they're in it for the long haul.
It's like a common law marriage for bros.
Christian: "Ya know Justin, we've been friends for over 10 years. I feel rather committed to this friendship."
Justin: "I feel like we have a bromage going on, man."