“We’re both Americans; we’re married,” Mr. Mize said. “We just found it really hard to believe that we could have a child that wouldn’t be able to be in our country.”

Their case illustrates the latest complication facing some families who use assisted reproductive technology, like surrogacy and in vitro fertilization, to have children. For years the techniques have set off provocative legal and ethical debates about what defines parenthood. Immigration and citizenship are the latest frontier in those debates.

At issue is a State Department policy, based on immigration law, that requires a child born abroad to have a biological connection to an American parent in order to receive citizenship at birth. That is generally not a problem when couples have babies the traditional way, but can prove tricky when only one spouse is the genetic parent.

The policy has come under intense scrutiny in recent months amid lawsuits arguing that the State Department discriminates against same-sex couples and their children by failing to recognize their marriages. Under the policy, the department classifies certain children born through assisted reproductive technology as “out of wedlock,” which triggers a higher bar for citizenship, even if the parents are legally married.

In one instance, a married Israeli-American gay couple had twin sons in Canada using sperm from each of the fathers. The biological son of the American received citizenship, but his brother, the biological son of the Israeli, did not. In February, a federal judge sided with the couple, calling the State Department’s interpretation of the immigration law “strained.” The department is appealing.