Relatives of the bomb victims expressed shock, anger and disappointment after the ruling. They said a cross-Border public inquiry was now the only hope of finding the truth behind the Omagh bombing.
Murphy, aged 52, from Ravensdale, Co Louth, will be freed pending a retrial.
He served three years of a 14-year sentence.
The judges ruled Murphy’s conviction, for aiding the bombers who carried out the August 1998 blast, was unsafe on two grounds.
It found the Special Criminal Court judges, Mr Justice Robert Barr, Judge Joseph Matthews and Judge Mary O’Halloran, did not fully analyse the possibility that other garda evidence might be tainted after accepting that notes from an interview were altered.
They were also wrong to take into account details of previous convictions.
Michael Gallagher, whose son Aidan died in the bombing, said it is “unforgivable” if judges did something they should not have done convicting Murphy.
Justice Minister Michael McDowell said he will study the judgment but declined to comment due to the retrial and criminal proceedings against two gardaí, who allegedly falsified interview notes and who are accused of lying to the original trial. The pair, detectives John Fahy and Liam Donnelly, appeared before Dublin District Court yesterday.
The appeal judges ruled the no-jury court:
Failed to bring the warranted extra degree of critical analysis to other, possibly contaminated, garda evidence.
Failed to assess the credibility of a third garda, Det Gda James Hanley, who took a key statement where Murphy admitted he knew his mobile phone was to be used to move bombs.
“Misdirected itself” by favouring scenarios for which there was no evidence. Took into account “manifestly inadmissible” evidence of previous convictions.
Mr Gallagher said he was in shock. “Thirty-one people died, including two unborn children, and not one person has been held to account after all the promises made by the two governments.”
Victor Barker, whose 12-year-old son James died after the bombing, has heavily criticised the investigation.
“I am just desperately disappointed,” he said.