“For two months after he left I did not receive any information about my son,” she said. “I didn’t know where he was.”

A tremendous sense of relief spread through the village after the BBC reported that a foreign navy had found a boat full of skeletal men clinging to life. A woman came to Mr. Rafiq’s mother and told her, “Your son survived, and my son survived,” she said.

Mr. Rafiq and the other survivors were brought to a refugee camp in Sri Lanka, where he said he was unable to speak for a month. He made a clawing motion at his throat, saying that it felt so dry that he could not make a sound. When he finally could talk, he asked to call his family. He called the one person in his village who had a cellphone.

Mr. Rafiq’s wife, Nur Hasina, 33, recalled the villager running to her with the phone.

“When I heard his voice,” Ms. Hasina said through an interpreter, “I asked him: ‘Where have you been? It has been too long. We have not heard anything from you.’ He said: ‘I cannot explain it right now. It’s a very long story. I will talk to you later, but I am alive and in Sri Lanka.’”

Mr. Rafiq and the other survivors refused to be sent back to Myanmar for fear of torture and execution. He stayed in Sri Lanka for four years, spending most of his time living in group housing provided by the Muslim Aid charity. After arriving in Dallas in 2013, he began working with the International Rescue Committee to reunite with his family.

His family had no legal rights to leave Myanmar, so they followed his instructions to sneak into India. They told border guards that they were traveling to Bangladesh for medical treatment, paid bribes and had other relatives furtively pass them their documents over a fence at the border. After two years in India, Mr. Rafiq’s mother, wife and three children arrived in Dallas in August 2016.