All crimes “motivated” by misogyny are to be recorded by police after the government bowed to pressure in the House of Lords to toughen the law on protecting women.

In a bid to avoid a government defeat, Home office minister Baroness Williams announced that all police forces in England and Wales would be asked from this autumn to record crimes caused by “hostility” based on sex or gender.

Campaigners hope the move is a key step towards wider change that would classify misogyny as a hate crime and give judges the power to impose tougher sentences for incidents of abuse and harassment, including street harassment.

The independent Law Commission watchdog has already recommended including the characteristic of sex or gender to existing hate crime laws. It has said the “vast majority of evidence” suggested several crimes are linked to misogyny.

Some seven police forces currently record misogyny as a hate crime but campaigners have called for a nationwide approach to help crack down on abuse and harassment of women and girls.

Baroness Williams said: “We do agree that data can be helpful. And we know that some police forces are already collecting it like Nottingham.

“So I will advise the House that, on an experimental basis, we will ask police forces to record and identify any crimes of violence against the person, including stalking and harassment and sexual offences, where the victim perceives it to have been motivated by a hostility based on their sex, which, as I have said, can then inform longer-term decisions.

“Once we have considered the Law Commission’s recommendations, we will shortly begin the consultation with the National Police Chiefs Council and forces on this, with a view to commencing the experimental collection of data from this autumn.”

Labour MP Stella Creasy, who has led calls for a change in the law and secured the commission’s review in 2018, welcomed the government’s move.

“I’m delighted that the government has listened to this cross-party and grassroots campaign to make misogyny a hate crime and is now taking the first steps towards making it happen.

“Recording where crimes are motivated by hatred of women will help us better understand the scale of the problem and so be better able to prevent these crimes – it should give all women confidence that if they come forward to report crimes they will be taken seriously, too.

“Now we want the government to implement the outcome of the law commission review in the sentencing bill so that our courts start to take misogyny and the crimes it drives seriously, too.”

The latest policy shift came amid widespread shock and anger at the death of Sarah Everard and the heavily criticised policing of a vigil in her memory last weekend.

Boris Johnson told MPs in prime minister’s questions that the country had to “address the fundamental issue of the casual, everyday sexism and apathy that fails to address the concerns of women”.

Crimes such as assault, harassment or criminal damage are already considered hate crime in cases where a person’s race, religion, sexual orientation, disability or transgender identity is involved, and are treated more seriously by the courts.

But campaigners have criticised the complex nature of the existing laws, and called for sex and gender to be added to the list.

Following the government concession, non-aligned peer Baroness Kennedy opted not to push to a vote her cross-party amendment to the Domestic Abuse Bill.

Kennedy had urged the government to “take a decision that will help ensure that all women everywhere can enjoy the same freedoms as men, when it comes to being able to go where we want and do what we want, without fear”.

“Since Sarah’s tragic murder came to the public attention, women everywhere have shared their stories of harassment, abuse, and violence at home and on the streets, and their frustration that all too frequently these crimes are not treated with the seriousness they deserve.”

Among backers of the amendment were Tory former ministers Baroness Altman and Lord Young.

The campaign has also been backed by metro mayors Andy Burnham, Steve Rotherham, Sadiq Khan and Dan Jarvis.

Earlier, peers inflicted a further defeat on the government in demanding that all domestic abuse victims receive protection and support regardless of immigration status and eligibility for public .

The Lords backed a cross-party change to the Domestic Abuse Bill, aimed at providing a blanket safeguard for women, by 310 votes to 232, majority 78.