The most common and most likely answer to this crossword clue is the 12 letter word COSMOPOLITAN
I've seen this crossword clue in The Guardian
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More clues leading to the result COSMOPOLITAN
- Company’s grouse about not entirely polite person, much-travelled?
- Worldly commander's grouse about petty officer settled
- Well-travelled commanders grouse about petty officer, drunk
- Sophisticated cocktail
- Sophisticated women’s magazine
- Worldly commander’s complaint about petty officer settled
- Sophisticated male with work in foreign locations
- Worldly Brits back in chilly island, brown
- Cultured and calm, holding small cleaning item I flog
- To complain so may be common all over the world
Buddy explains!

The straight part
We know that "Well-travelled" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case well-travelled = cosmopolitan.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 COSMOPOLITAN mean?
COSMOPOLITAN noun- A cosmopolitan person; a cosmopolite.
- A cocktail containing vodka, triple sec, lime juice and cranberry juice.
- A butterfly, Vanessa cardui
- Inclusive; affecting the whole world.
- (of a place or institution) composed of people from all over the world.
- (of a person) at ease in any part of the world; familiar with many cultures.
- Growing or living in many parts of the world; widely distributed.