My brother's father-in-law is my what?

Title says it all, really. Is there a word for this relationship?

Yes. He would be your brother’s father-in-law.

That’s four words.

In English, no. In other languages, possibly.

My husband calls them “shirttail relatives.”

By them I mean your brother’s father-in-law, and other not-quite-relatives that you feel related to because you are both related to the same people in different ways. I think it’s just something that he made up himself, but I kind of like it.

In my case, that word would be “stranger.”

I literally don’t even know if my brother’s father-in-law is alive. Never met the man as far as I know.

“Absolutely nothing.”

I’ve wished for a word/term too. It’s the same thing with trying to refer to my wife’s sister-in-law. As mentioned by others, there is no English term for the relationship.

“Acquaintance”.

Your forgot the [Dark Helmet] tags.

“shirttail relatives” is a good term for such people, but if your husband made it up independently, he’s not the first or only one to do so. I’ve heard and read it in many places, though I can’t tell you how common or widespread it is.

My family refers to such relations as “uncle-ish,” which is decidedly useless.

I’ve been known to refer to my BIL’s brother (a.k.a. my sister’s BIL) as my “brother-in-law-in-law.”

You’re brother-in-law’s brother would also be your brother-in-law.
Your sister’s brother-in-law would be different.

He’s your sister in law’s father, you nitwit!

Tris

I’m still tossing up between “complete and utter stranger” and “who cares” because I have used both interchangeably.

My wife calls her sister-in-law “horse face.”

A term I use, somewhat tongue in cheek, for my brother’s wife’s sister is sister-in-law once removed.

We call my sister in law’s husband Jerry.

Sorry, I read that as “still tossing off” and got a very, very strange visual.

That sounds good. From now on we will call all of them Jerry.