A would-be brother-in-law who is in a commited relationship with your
sister but isn't legally married to her.
These relatives are awkward to introduce because they're not necessarily your friends, but technically not your relatives either. You don't want to insult him by going into detail about his relationship with your
sister in public, but if you call him your brother-in-law, awkwardness might arise when
people glance at the non-existent wedding ring.
This also applies to would-be parents and sisters-in-law.
Jack: Hey
Ted, why don't you invite your
brother-in-theory onto our softball team?
Ted: Are you kidding? Then my father-in-theory's gonna want to join too. There's a reason I didn't marry into that crazy family, ya know.