During WW2, An american serviceman was caught murdering a woman in Melbourne, and it was discovered that he had killed several women on Australian soil. Australian law was not recognised, and bypassed, and the US Military had tried the man, even though he was still on Australian soil.
Not quite accurate. The man was a rapist, not a murderer. Australia had first dibs on him, due to the crime occuring on Australian soil, but chose to surrender him to the US military. The reason? Australia, at the time imposed the death penalty only for murder, while the US military had (and still does have, I believe) the DP for rape. So, the Australian government allowed the US military to try him by Court Martial, and he was taken off to one of the US islands and hung.

There was a LOT of controversy about it in Australia at the time, but it was primarily aimed at the Attorney General, whom it was felt had erred in handing the serviceman over.