A woman has lost a court of appeal challenge over her name being removed from a child’s birth certificate after her ex-wife admitted she secretly had sex with their sperm donor.

The “unprecedented” and “unusual” case centred on the question of who were the legal parents of a girl, now aged six.

The child, referred to as X, was born amid an “informal conception arrangement” between two women and a man they met through an online advert, judges were told.

The couple, referred to as P and Q, met the man, F, for the first time in a pub in late 2016, hoping to find a sperm donor to impregnate P.

The women “formed a favourable impression of him” and the three of them then signed a sperm donor agreement, agreeing to use artificial insemination.

When two artificial attempts failed, the court heard how P contacted F for support and the pair had sex. P had been “very upset and depressed” about the lack of success and arranged to meet him at her parents’ house while they were abroad. She found him “friendly and sympathetic” and initiated the intercourse, a court was told.

P and F had sex three times at her parents’ house, unknown to Q, with the third time coinciding with a third artificial insemination attempt. Judges said it was “impossible to know which method of insemination led to [the child’s] conception”.

P and Q later divorced and, amid disagreement over the care of the child, P revealed the truth about her encounter with their donor, and secured a court declaration earlier this year that F was the child’s legal parent.

In her ruling in April, Mrs Justice Knowles said the case was “a cautionary tale about the consequences for a child and for a same-sex couple of both deceit as to how that child came to be conceived and the unreliability of informal arrangements for artificial insemination”. She added: “The fallout from this couple’s separation has been devastating for each of them and for their named sperm donor.”

She concluded that, on the balance of probabilities, while Q had not consented to sex between P and F, the method of the child’s conception was “unclear”.