The most common and most likely answer to this crossword clue is the 14 letter word DOUBLE-CHINNED
I've seen this crossword clue in Independent
D
O
U
B
L
E
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C
H
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N
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D
Buddy explains!
The straight part
We know that "Greeted Anglo-Chinese style, say,/showing some weightiness?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greeted anglo-chinese style, say,/showing some weightiness? = double-chinned.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.