The most common and most likely answer to this crossword clue is the 5 plus 10 letter word STATE DEPARTMENT
I've seen this crossword clue in The Daily Telegraph
S
T
A
T
E
D
E
P
A
R
T
M
E
N
T
Buddy explains!
The straight part
We know that "branch of US government" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case branch of us government = state department.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.