The celebrities have launched a petition calling for the government to change the law to “protect the reputations of all innocent suspects, whether well-known or not, from the lasting stigma of a false sexual allegation”.
It passed 10,000 signatures on Monday afternoon and will receive a government response, while 100,000 would trigger a parliamentary debate.
In an open letter to Sir Cliff and Mr Gambaccini, the End Violence Against Women Coalition said false allegations were no higher for sexual offences than other crimes and “our big problem is the enormous reluctance victims have to report” attacks.
Only 10-15 per cent of rapes are estimated to be reported to police, and only 1.7 per cent of those result in a charge in England and Wales.
Speaking at a launch event in Westminster, Sir Cliff said: “We have both been through the mill. When you know you didn’t do it, you feel you’re in a hole you can’t get out of.”
The 78-year-old singer won a privacy case against the BBC over its coverage of a South Yorkshire Police raid on his home in August 2014, following a child sex assault allegation that he denied.
Sir Cliff was never arrested, and in June 2016 prosecutors announced that he would face no charges.
He said he didn’t sleep properly for four years, came out in shingles and felt he had been “hung out to dry”.
Mr Gambaccini, a radio DJ, was arrested in October 2013 as part of Operation Yewtree, set up in the wake of the revelations about paedophile Jimmy Savile.
The 70-year-old, a regular fixture on the airwaves for decades, spent a year on bail before the case was dropped.
Mr Gambaccini said he felt ”betrayed“ by law enforcement agencies and claimed there was a “false allegation crisis” in Britain.
“At the moment accusers have lifelong anonymity and the accused have no seconds of anonymity,” he told BBC Radio 4’s Today programme. “And this does, unfortunately, encourage everyone from liars to lunatics to make some false accusations and get in on the action.”
But the College of Policing said it cannot “prevent the media relying on information from sources outside the police in order to confirm identities”.
Sir Cliff and Mr Gambaccini are working with a group called Falsely Accused Individuals for Reform (Fair), which includes Lord Janner’s son – Daniel Janner QC – former Tory MP Harvey Proctor and acquitted rape suspect Liam Allan.
In an open letter, a group representing victims called on them to reconsider their campaign and “prevent significant harm to survivors of abuse across the UK and the legal process”.
“The harm that those who are accused and acquitted feel is, we believe, a result not of the fact of being named but of terrible media representation of sexual violence cases, accompanied by a collective failure to uphold the presumption of innocence,” wrote Sarah Green and Rachel Krys, co-directors of the End Violence Against Women Coalition.
“We should all stand up and tackle these problems, but it is of the utmost importance that police retain the ability to name a suspect when they have good investigative reasons to do so.”
The letter questioned why anonymity should be legally guaranteed before charge for sexual offences, but not for other severe crimes such as murder.
Campaigners cited the cases of Rolf Harris and Stuart Hall, when witnesses came forward after reading about other accusations.
Mr Gambaccini accepted that publicity surrounding suspects can encourage accusers to come forward but said: “If there is a strong case, as there was, for example, in the taxi driver case [John Worboys] ... that will come to charge and at that moment other people can come forward.”
Research by the Crown Prosecution Service (CPS) found that in a period where more than 5,500 rape cases were prosecuted, only 35 people were prosecuted for making a false allegation, which was no higher for sexual offences than for any other crime type.
A 2012 parliamentary research briefing said people who allege they are the victims of certain sexual offences have been automatically entitled to lifelong anonymity since 1976 but the same right was stripped from defendants in 1988.
In 2010, the newly-elected Coalition government indicated that it would “extend anonymity in rape cases to defendants” but the proposal was dropped because of insufficient empirical evidence.
A Ministry of Justice report questioned “whether the inability to publicise a person’s identity will prevent further witnesses from coming forward, or further unknown offences by the same person from coming to light”.
Deputy Chief Constable Janette McCormick, from the College of Policing, said: “Police balance the need to protect the public with an individual’s right to privacy and we do not name those arrested, or suspected of a crime, unless there are exceptional circumstances to do so. These include a threat to life or to warn the public about a dangerous or wanted individual.
“If a police force wants to name an arrested person before they are charged it requires approval by a chief officer following consultation with the CPS.
“This approach is clear in our national guidance and is followed by police forces across England and Wales.”
The government has not confirmed whether it is considering any formal review or legal change.
A Home Office spokesperson said: “The government believes that there should, in general, be a right to anonymity before the point of charge. But there will be circumstances in which the public interest means that an arrested suspect should be named.”