The most common and most likely answer to this crossword clue is the 13 letter word REMONSTRATION
I've seen this crossword clue in Financial Times
R
E
M
O
N
S
T
R
A
T
I
O
N
More clues leading to the result REMONSTRATION
Buddy explains!
The straight part
We know that "Protest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case protest = remonstration.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.