SUNNYVALE, Calif. — Eddie Oh, an industrial engineer, lost his job during the financial crisis that gripped South Korea in 1998. With no prospects, he scrounged together his savings to pay his family’s airfare to California. They were going on vacation, he told the United States embassy, which issued six-month visitor visas for the family.
The Ohs headed to Sunnyvale, a middle-class community in California’s Silicon Valley where a relative already had rented a small apartment. The Ohs moved in, nine people crammed into two rooms. Mr. Oh got to work painting houses. His wife found a job as a waitress. And their children, Eli, 11, and Sue, 9, started school.
“We were constantly in debt. We struggled to pay the rent,” said Eli Oh, who grew up to be a critical-care response nurse at Stanford University. “Nobody ever thought we were illegally here because we didn’t fit the stereotype.”
They are hardly alone. Though President Trump has staked much of his presidency on halting the movement of undocumented immigrants across the southern border, the Oh family’s roundabout route to residence in the United States is part of one of America’s least widely known immigration stories.
Some 350,000 travelers arrive by air in the United States each day. From Asia, South America and Africa, they come mostly with visas allowing them to tour, study, do business or attend a conference for an authorized period of time. But when they stay beyond when their visas expire, some of them fall into the same illegal status often associated with migrants showing up at the border.
Nearly half of the estimated 11 million undocumented immigrants now in the country did not trek through the desert or wade across the Rio Grande to enter the country; they flew in with a visa, passed inspection at the airport — and stayed.
Of the roughly 3.5 million undocumented immigrants who entered the country between 2010 and 2017, 65 percent arrived with full permission stamped into their passports, according to new figures compiled by the Center for Migration Studies, a nonpartisan think tank. During that period, more overstayers arrived from India than from any other country.
“A big overlooked immigration story is that twice as many people came in with a visa than came across the border illegally in recent years,” said Robert Warren, the demographer who calculated overstay estimates by using the Census Bureau’s annual American Community Survey and shared those figures with The New York Times.
As Mr. Trump has called for hiring thousands of new Border Patrol agents and erecting miles of new fencing, federal immigration authorities have devoted relatively few resources toward the much larger numbers of undocumented immigrants who have overstayed their visas.
The Department of Homeland Security said it has succeeded in bringing the number of visa overstayers down slightly over the past two years, but enforcement is difficult because authorities are only beginning to gain access to better data on who has and has not flown out of the country.
“Once they are in the country, they are home free because there is so little interior enforcement,” said Jessica Vaughan, a former federal visa officer who is now policy director at the Center for Immigration Studies, which lobbies for restricting immigration.
Nearly half of the undocumented population overstayed a visa.
Overstayers represent about 46 percent of the 10.7 million undocumented immigrants in the United States, according to the migration center’s data. This is not necessarily because of a huge jump in the number of people overstaying their visas; rather, their proportion of the undocumented population has soared amid a huge decline in border crossings since 2000.
The largest number of overstayers — about 1 million — hail from Mexico, a neighboring country with a long history of commercial and family ties and substantial flows of people across the border. But the picture is changing. Between 2010 and 2017, 330,000 Indians overstayed their visas, more than from any other country. Large numbers of people from China, Venezuela, the Philippines, Brazil and Colombia also overstayed.
Many undocumented Asians — including a large number from India — have settled like the Ohs in and around Sunnyvale, about 50 miles southeast of San Francisco, according to the Center for Migration Studies analysis.
Apple, LinkedIn and other tech titans in the area employ many whom the companies have sponsored for legal work visas or permanent residency in the United States.
Some of them stay on as independent programming contractors after their visas have expired or after leaving a company that sponsored them for a visa.
But they are only part of the story. Many undocumented Indians here in Sunnyvale have low-skilled service jobs, catering to their well-heeled brethren who frequent the Indian supermarkets, eateries and clothing shops that line El Camino Real, the main commercial corridor.
They are people like S. Singh, 24, who works at a diner where the Indian lunch crowd on a recent afternoon dined on spicy lentils and spinach with flatbread and sipped masala chai. Mr. Singh, who like most others interviewed for this story declined to share his full name, said that he had arrived as a tourist two years ago.
At Indian grocery stores nearby, Indian workers arranged shelves stocked with Taj Mahal tea, basmati rice and canned Kesar mangoes. They hesitated to answer questions, beyond saying that they had entered as tourists. One of them said that he had come on a student visa that had expired.
Inside a closed restaurant on a recent afternoon, two Indian men and two Indian women workers slept before dinner service, their bodies draped over a long bench where patrons would later be seated. Ankit, an Indian engineer on a work visa who had hoped to grab a bite, only to realize it was too early, surmised that they were undocumented — just like the Indian Uber driver who had brought him there.
“There are no legal pathways for people working in restaurants and grocery stores,” he said. “These workers are coming for a better life.”
Tracking visa overstayers is difficult.
The government reported that nearly 670,000 travelers who arrived by air or sea and were supposed to depart in the 2018 fiscal year had not left by Sept. 30, 2018. That number had dropped to nearly 415,700 by March 2019, because many people overstay by just a few months.
But developing policies to curb overstays requires accurate data, experts say, and Homeland Security officials still lack a reliable system to track them.
Most travelers are photographed and fingerprinted at American consulates abroad when they receive a visa and then again on arrival in the United States. But Customs and Border Protection still depends overwhelmingly on biographical information from the manifests of departing travelers, provided by airlines, to tally who did not leave in time, or at all.
In 2016, federal officials began working with airlines and airport authorities to install a biometric facial-comparison system at departure gates. A digital picture taken of those boarding a plane to leave the country is compared to the one taken on their arrival.
Thus far, the program covers 4 to 5 percent of those departing by air each day, said John F. Wagner, a deputy assistant executive commissioner for Customs and Border Protection. He said in an interview that his agency hopes to cover 90 percent of departing travelers within three years.
Immigration and Customs Enforcement, which enforces immigration rules in the interior of the country, said that it puts a priority on identifying those who pose potential national security or public-safety threats. In fiscal 2018, its Homeland Security Investigations unit made 1,808 arrests in connection with visa-violation leads.
Many do not mean to lose their legal status.
Many of those who overstay their visas do not intend to stay illegally, said Kalpana Peddibhotla, an immigration lawyer in the San Francisco Bay Area.
“They entered with a specific purpose and fell out of status for a variety of reasons, only to realize there is no easy mechanism to correct their status violations,” she said.
Graduates of American universities, allowed to remain in the United States for a period of time to work, run afoul of deadlines or commit errors on immigration forms that automatically render them deportable. Sometimes employers transfer foreign workers to a new site and fail to amend their paperwork, as required, which also cancels their legal status.
“They stay because they built their lives here, bought homes here, had children here,” said Ms. Peddibhotla, who became familiar with Indian overstay cases while on the board of the South Asian Bar Association of North America.
Among Asians, in particular, being undocumented brings shame to the family. Like several others, the senior Mr. Oh, his wife and daughter, declined to be interviewed for this article even though the daughter was able to help her parents obtain green cards after she married an American.
“My parents are not proud of breaking the law,” said Mr. Oh, who also now has a green card. “To this day, most of their church friends do not know they were undocumented.”
In places like Sunnyvale, it is not hard for people to disguise their immigration status.
“Especially if they’re not from Mexico or Latin America, no one suspects them of being undocumented,” said Kathy Gin, executive director of Immigrants Rising, a San Francisco-based advocacy organization that works with undocumented youth.
“Their parents encourage them to keep their heads low, not share their stories, not speak out about immigration issues,” said Ms. Gin.
Marilyn Omatang left Manila in 2004 with her eldest child, Dean, then 12, to join her husband, who had arrived in California two years earlier. “She told me we are going to Disneyland,” recalled Dean, which they did.
But they had to remain in the United States to earn the money to provide for a special-needs child who required pricey medical care and for the schooling of three others, all living in the Philippines with relatives.
“With just a job at a 7-Eleven, I could pay for the medical treatment and their education,” said Ms. Omatang, 56, who rose to the position of manager, only to be let go after a co-worker reported to their boss that she was undocumented.
For more than a decade since then, she has been a caretaker to wealthy seniors in Silicon Valley, never revealing her status, and using a different name to report her income to the federal tax authorities.
One of her employers, Ms. Omatang said, is a Trump supporter who favors a tough approach to undocumented immigration.
“I heard her say, ‘just send all those illegal people back home,’” she said. “And I thought, ‘Oh, oh. If you only knew.’”