Careers: Military--Army, Navy, Airforce, Marines, Coast Guard/A bit of a complicated question ...
Well ... this is a question for a story. It's a bit complicated so I hope I can explain properly :D It seems your experience that you listed might be well suited to answer this ;)
First off, this takes place in an entirely fictional setup that I did not create all of ... but things are based almost *directly* off of the current US Navy (ok, I know you're in the Coast Guard)
Essentially, it's as if there is the UN, and the naval force then is part of the *UN* and not part of the individual member countries ...
(The analogy does not quite hold up when it is the current setup of federal govt. vs. state govt. and laws.)
All people are citizens of their country and the "UN" and are of course subject to the laws of both. The "UN" upholds its members' laws of course but does not have the jurisdiction to enforce laws that are not its own. (Yes, just like federal vs. state.)
A guy, an officer in this, commits a crime which is only against the laws of his own member country (which he retains citizenship to), while on board ship ... the victim is not a citizen of the same place he is. Due to several irrelevant factors, she chooses not only to let the entire matter go, not to press charges or pursue any type of investigation or *anything*, but also will not identify him as the one who did it ...
My question is: what legal authority does the ship's commander have in this situation? He is aware of the guy's country's law, at least. Can he press for an investigation or conduct an investigation based on that he knows this is a crime in another jurisdiction? Can he be a part of or conduct an investigation at all, regardless of jurisdiction?
Does it make a difference that the victim is being so ... unhelpful? :D
I hope I haven't left out any major information!
I think your question is a matter of jurisdiction. There are three components of jurisdiction (persons, location and substantive law). For example as a federal law enforcement officer I have jurisdiction over people (example us citizens), in certain areas (example us territorial sea), for the enforcement of certain laws (example federal fisheries laws...of note I can't enforce state fisheries laws).
In your situation the CAPT would have jurisdiction over his(or her) crew, the ship but the law broken would need to be one that he(or she) is permitted to enforce.
In reality we have UN naval task forces that serve under the TACTICAL control of UN commands but remain under the ADMINISTRATIVE control of there national fleet. Rarely if ever do you have crewmembers from differing nations aboard one ship for really the very situation you propose. Certainly you can have liaison officers attached to a unit but actual crewmembers are usually citizens of the same country subject to the same laws.
LCDR John Fiorentine USCG