BEIJING (Reuters) - China said on Thursday that it was coordinating with the U.S. navy in the search for a missing U.S. sailor in the South China Sea, a rare show of goodwill between the navies in the disputed waters.

The U.S. 7th Fleet said on Tuesday that U.S. and Japanese ships were looking for an unnamed sailor from the USS Stethem destroyer who had gone missing during a routine operation in an unspecified section of the South China Sea.

It said multiple searches were conducted inside the ship, but to no avail.

China's Defence Ministry said in a statement that its Liuzhou guided-missile frigate was in nearby waters conducting war-readiness duties, and "on the basis of humanitarian spirit, and according to the code for unplanned encounters at sea, carried out operational coordination with the U.S. side".

It said the sailor went missing when the U.S. ship had been more than 100 nautical miles southwest of the contested Scarborough Shoal, but did not elaborate.

China claims nearly all of the South China Sea, through which about $5 trillion in ship-borne trade passes each year, a stance contested by Brunei, Malaysia, the Philippines, Taiwan and Vietnam.

Washington has criticized Beijing's construction of islands and build-up of military facilities there, concerned they could be used to restrict free movement and extend China's strategic reach.

But Chinese officials say that U.S. "freedom of navigation" operations in the waters violate China's sovereignty and raise tensions in the region.

China has said one of the reasons for its island building is to better meet its humanitarian and search and rescue obligations at sea.

In June, the U.S. navy said a sailor had been found alive aboard the USS Shiloh after U.S. and Japanese vessels spent 50 hours searching thousands of square miles of waters off the Philippines.

(Reporting by Michael Martina; Editing by Nick Macfie)