The discovery mirrors a storyline in teen soap Hollyoaks where a brother and sister slept with each other without knowing they were related
It was only after they married, according to Lord Alton, that they became aware of the "appalling" truth and sought to have their union dissolved. Marriages can be annulled in cases where there is a "prohibited degree of consanguinity" - blood relationship.
The case came before a judge, sitting in private, in the High Court Family Division where, according to Lord Alton, "he had to deal with the consequences of the marriage that they entered into and all the issues of their separation".
No other details about the couple, their relationship, or the annulment have been made public but the peer added: "It was appalling for this couple to discover they were married to a close relative.
"And I believe the Government will leave itself open to class actions in the future if it collaborates in keeping information of this kind from children who have been donor-conceived."
Lord Alton and his supporters want the genetic history of a child recorded on its birth certificate.
The matter will be debated next week at the next reading of the controversial bill.
He said: "One of its provisions is to deny the child's right to know about its biological identity.
"This means the state is colluding in a deception. The Government has drawn up legislation that will deny you this knowledge until you are 18. But you could be married by then, or have fallen in love with someone.
"This will lead to these heartbreaking situations.
"If you start trying to conceal someone's identity, sooner or later the truth will out.
"And if you don't know you are biologically related to someone, you may become attracted to them and tragedies like this may occur."
Audrey Sandbank, a family therapist and author of Twins and Triplet Psychology, said the twins are likely to have felt "like soulmates" when they met for the first time because of their shared genes.
"This is a terrible trauma for them. They lost each other as babies and now they have lost each other again. They have been bereaved twice."
Pam Hodgkins, chief executive of the charity Adults Affected by Adoption, said there had been previous cases of separated siblings being attracted to each other.
"We have a resistance, a very strong incest taboo where we are aware that someone is a biological relative. But when we are unaware of that relationship, we are naturally drawn to people who are quite similar to ourselves.
"And of course there is unlikely to be anyone more similar to any individual than their sibling."
Mo O'Reilly, director of child placement for the British Association for Adoption and Fostering, said such an incident was traumatic but very rare.
"Thirty or 40 years ago, it would have been more likely that twins be separated and brought up without knowledge of each other."
But she claimed that there was far more openness today with adopted children about their birth parents and background.
Last year, a German brother and sister who were separated as children and fell in love after being reunited campaigned for the right to continue their relationship. Patrick Stubing was jailed for two years for incest after he had four children with Susan, who by then he knew to be his sister.
The 29-year-old locksmith was given up for adoption as a baby and did not meet his real mother or sister until he was 18.
The couple, who live near Leipzig, claim they fell for each other when their mother died and are fighting to have the laws regarding incest changed.