The most common and most likely answer to this crossword clue is the 13 letter word CLANDESTINELY
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More clues leading to the result CLANDESTINELY
Buddy explains!

The straight part
We know that "happens on the sly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case happens on the sly = clandestinely.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.
What does CLANDESTINELY mean?
CLANDESTINELY adverb- In a secret or covert manner.
- In an illicit way, i.e. not permitted by the law or regulations.