The most common and most likely answer to this crossword clue is the 8 letter word MALDIVES
I've seen this crossword clue in The Daily Telegraph
M
A
L
D
I
V
E
S
More clues leading to the result MALDIVES
- Short man plunges into water in tropical isles
- Islands emphatically shut up about what some people do there
- Clubs overcome by disorder in French archipelago
- Beat up rich man in island republic
- Holiday islands will be hit over disreputable bars
- Mum enjoys life touring round 500 islands
- Countrymen accept 50% share
- Countrymen get 50% share
- Islands where men welcome divorce
Buddy explains!

The straight part
We know that "island republic" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case island republic = maldives.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.