Last summer, the police in the Fijian capital of Suva closed their investigation into the shootings. They reasoned that the incident did not occur in their national waters, nor did it involve their vessels. Since no Fijian mariners had been reported missing, they concluded none of their citizens were among the victims.

When governments investigate incidents like this, their goal is typically not to find the culprit, said Glen Forbes from OceanusLive, the maritime risk firm. “It’s to clear their name.”

The video, which includes people speaking Chinese, Indonesian and Vietnamese languages, shows three large vessels circling the floating men. A banner that says “Safety is No. 1” in Chinese hangs in the background on the deck of one of the ships. A fourth vessel, which maritime records indicate is a 725-ton Taiwanese-owned tuna longliner called Chun I 217, passes by in the background.

Lin Yu-chih, the owner of the Chun I 217, which remains at sea, said that he did not know whether any of the more than a dozen other ships he owns or operates were present when the men were shot. “Our captain left as soon as possible,” Mr. Lin said, referring to the shooting scene.

Though the date of the shooting is unknown, he said that he believed it occurred in 2013 in the Indian Ocean, where the Chun I 217 has been sailing for the last five years.

Mr. Lin declined to release any details about the crew of the Chun I 217 or the report he said he asked the captain to write about the killings after the Taiwan police contacted his company. Mr. Lin, a board member of the Taiwanese tuna longliners association, said the private security guards on his ships were provided by a Sri Lankan company, which he declined to name. The Taiwan prosecutor’s office, which is looking into the matter, declined to comment.

With one of the world’s largest tuna fleets, Taiwan’s fishing industry is among the nation’s biggest employers and most politically powerful sectors.