They do not know whether their own embryo was born alive and switched in the nursery and, if so, who is raising it.
They have not been able to identify the genetic parents of "X", the baby they have raised, as the clinic has not revealed its records.
The couple's lawyer, Seonaid Abernethy, told the Listener's Catherine Woulfe that without any genetic records, Baby X was stateless, anonymous, a "scrap of humanity" with no passport and no identity.
The matter went to the Family Court, where Judge David Burns said: "The situation is one of urgent humanitarian need ... [The baby] has a right to an identity, a family, and to grow up in a family environment of love and happiness.
"That is able to be provided by Mr and Mrs Y in the absence of anyone else and this protective umbrella is currently under great pressure."
The couple want to know what happened to their embryo. But they also want to adopt the toddler they have brought up as their own, and fear that if they fail in their legal bid, the child will end up in a foreign orphanage. The couple hope the foreign adoption orders will come through in about six months.
Listener editor Pamela Stirling said the magazine has taken great care to avoid jeopardising the adoption process by taking care about what details it published.
• Full story in the NZ Listener, on news-stands tomorrow.