The most common and most likely answer to this crossword clue is the 13 letter word STREET-SMARTS
S
T
R
E
E
T
-
S
M
A
R
T
S
Buddy explains!
The straight part
We know that "wily Chicago-style know-how" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case wily chicago-style know-how = street-smarts.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.