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1.
A conjunction of 'bro' and 'mission,' usually in reference to a mission of bros (e.g., a road trip).

Subtext is similar to the portmanteau 'bromance.'
"Hey Hermano, let's go on a bromission to Redmond."

"Roger that. I'll pick you up in 5, Hermano out."
by phooji May 17, 2010
 
2.
When you (male) ask your guy friend for permission to hook up with, bang, or date your friend's ex-girlfriend, ex-wife, girlfriend, wife, or pretty much any girl your friend has been with.
"Damn, Joe's ex-wife is all up on my junk."

"Dude, you'd better call Joe and get bromission before you hit that."
by Camp Custy March 17, 2009
 
3.
permission given to a guy by another who is considered a "bro" to date or pursue a girl that the "bro" already likes!
A girl you like likes your friend. He thinks she is cute so you give him the bromission to pursue her realizing she isnt interested in you.
by Kinser Crissman April 23, 2013