The most common and most likely answer to this crossword clue is the 8 plus 3 plus 6 letter word TRINIDAD AND TOBAGO
I've seen this crossword clue in The Times
T
R
I
N
I
D
A
D
A
N
D
T
O
B
A
G
O
Buddy explains!
The straight part
We know that "Caribbean state" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case caribbean state = trinidad and tobago.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.