For fact checkers themselves, the work takes a toll. Members of Rappler’s staff have received death and rape threats. Rappler brought in a psychologist. It debated bulletproofing the windows and installed a second security guard.

The job also requires patience. One busy day this summer, the newsroom’s fact-checking team asked Mr. Esmaquel, who covers religion, to look into the story about Mr. Duterte’s debating the bishop. Even though the story had been shared nearly 4,000 times and had reached more than a million followers, he knew right away that it was a hoax. But he still had to call up the Archdiocese of Manila for comment.

“I said, ‘Father, I know that this is fake, but I need a quote from you,’” he said.

This kind of work doesn’t end for Mr. Esmaquel, 32, and his colleagues.

“We kill one,” he said, “and another one crops up.”

Rappler has experience battling misinformation. It was founded as a scrappy investigative reporting and entertainment outlet in 2012 by Maria Ressa, a former CNN bureau chief in Manila and Jakarta, Indonesia. She persuaded three female friends — a group she nicknamed “the manangs,” a Tagalog word for old ladies or sisters — to leave their high-powered jobs at news stations and magazines. They shared an optimism that the internet would be a platform for the powerless to find a voice and that Rappler, which is a hybrid of “rap” and “ripple,” would be a vehicle for social change.

Instead, the internet in the Philippines became an outlet for threats and deceit.

That is particularly true on Facebook, through which about 97 percent of internet users in the Philippines get access to the web. Before the Philippine election in May 2016, fake accounts appeared on Facebook spreading positive stories about Mr. Duterte, who was running for president as a blunt-spoken, antidrug populist. They also excoriated Mr. Duterte’s opponents, often with personal and inflammatory attacks. Much of the content was untrue.

After he won the election, Mr. Duterte waged an antidrug campaign that has led to thousands of deaths, provoking an international outcry. Many of his critics and political opponents, including Rappler, have run into legal problems. Facebook campaigns have underpinned much of this activity. At times, officials in Mr. Duterte’s administration have openly shared misinformation on the platform.