The most common and most likely answer to this crossword clue is the 8 letter word TAPENADE
I've seen this crossword clue in The Guardian
T
A
P
E
N
A
D
E
Buddy explains!

The straight part
We know that "culinary paste" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case culinary paste = tapenade.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 TAPENADE mean?
TAPENADE noun- A Provençal dish consisting of puréed or finely chopped olives, capers, anchovies, and olive oil, usually eaten with bread as an hors d'œuvre.