“I didn’t know until we were in the air,” said Carol Montgomery, 67, a retired administrative assistant from San Clemente, Calif. “I saw an area of plastic sheeting and tape.”

While the planes were aloft, the State Department and the Department of Health and Human Services said in a joint statement that the results for 14 passengers who had been tested two or three days earlier had come back positive — just as they were heading to the airport in buses.

American officials, it turned out, had started bringing the passengers home without actually knowing their test results. But because the evacuation had already begun by the time Japanese officials relayed those results, officials decided to let the infected evacuees, who were not yet exhibiting symptoms, board the planes and sit in the back, separated from other passengers by plastic sheets about 10 feet tall.