Three-year-old Binh and Phuoc suffer from a rare life-threatening condition and each desperately need a liver transplant.
Their dad, Michael Wagner, from Ontario, is a match for the girls – but he can only donate part of his liver to one of them.
Michael and his wife Johanne adopted the twins when they were just 18 months old after seeing they were in poor health at a Vietnamese orphanage.
Both of them suffer from Alagille syndrome - a genetic disorder that affects the liver, heart and other organs.
Their liver functions are slowly coming to a halt - meaning a transplant could potentially save their lives.
However they will need two donors to save them both - so have begun a search for another match.
They have started a Facebook page in a desperate bid to find another donor.
While the hunt continues, Michael will work with a team of doctors at Toronto General Hospital to decide which of the girls will receive the transplant.
Johanne told The Canadian Press: "We told them we didn't want to be burdened with the decision making."
Doctors are expected to perform the surgery in around two weeks, but the family are yet to find a second donor.
Johanne added: "When we saw them at the orphanage we were shocked, really.
"They were 9 pounds at 18 months. So we left the orphanage that day and went to buy two little containers with dragonflies on them and that's what we were going to lay their ashes in if they didn't make it."
Despite the health problems, Johanne has no regrets.
She said: "We would travel this path all over again. They have taught us openness, they have taught our children sharing and openness. It's been nothing but a wonderful mess."
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