by Ashley Dunn

About the author: Ashley Dunn writes for the New York Times.

In the comfort of a KLM flight from Belfast, Francis and Theresa S. arrived in the United States in October 1994 as tourists. But they had no intention of returning to their home in Northern Ireland after their sightseeing in New York was over. They were coming to stay on as illegal immigrants.

Unlike the thousands of Mexicans and Central Americans clambering over the flood-lit fences of the southern border, the couple merely walked through an inspection booth at Kennedy International Airport. An immigration official, checking their passports and plane tickets, smiled and waved them in.

“It was that easy,” said Francis, who along with his wife did not want to be fully identified for fear of being uncovered. “They just stamped...