This week, sexist and racist trolls have borrowed the tactics of the CIA and Scotland Yard and sent in agents provocateurs to spread disharmony among online activists. Using stock photos and the stolen information of real activists, users of sites such as 4chan started hashtags including #Endfathersday and #whitescantberaped that are deliberately designed to provoke sections of the social justice left into internal arguments.

Most of these trolls have posed as women of colour, whom they call "black bitches". Other users have falsified racist tweets from prominent feminists and leftists, and created sock-puppet accounts to make sure the fake tweets are seen and condemned. Users including Shafiqah Hudson picked up on the scam, and identified at least 200 such accounts. Someone has gone to considerable effort to pull off a swindle intended to exploit the "weaknesses" of a movement that, despite a tendency to turn on itself, is growing in strength.

In the past five years, Twitter and other social media platforms have become key sites of cultural activism. From feminism to anti-racism and transgender rights, people have used microblogging sites to make their experiences visible, share stories of injustice, organise collectively and educate themselves. Hashtags such as #solidarityisforwhitewomen, created by Mikki Kendall, and #yesallwomen, created by user @gildedspine, have been enormously effective at making sure sexism and racism, including within the left, can no longer be ignored – at least, not if you spend as much time online as the authors of this piece do.

A network is only as good as the bonds of trust upon which it is built. That is why the agent provocateur is such an effective form of sabotage – online and offline. It's not just the false information they spread, but the atmosphere of suspicion they foster. If anybody could be a plant, nobody can truly be trusted, and friends become paranoid and turn on one another.

Of course, radicals and progressives have never needed much encouragement to cannibalise our own passions. Infighting is a feature rather than a bug of social justice movements, though it is hardly unique to the left, and if we wait for everyone to get along before we start to create change, we're going to be waiting a very long time. Existing tensions, however, are vulnerable to exploitation – and these trolls have hit the weak nerve of the modern feminist movement, capitalising on the often troubling ways it has dealt with criticism from constituencies of women considered marginal. Our enemies know us better, perhaps, than some of us hoped.

It's common to hear valid critique dismissed as unconstructive "infighting" that will only derail us from a universal vision of female empowerment. Twitter can be a bitter place – especially in arguments where space is limited, empathy is difficult to hang on to, and everybody involved cares so much. Sometimes, things can get messy. Personal abuse is rarely constructive, but heated emotions are understandable. Our movement is made up of many voices, and we cannot and should not present ourselves as a homogenous lump.

For decades, attempts to undermine feminism, socialism and anti-racism have focused on our tendencies to find fault among ourselves. That's just one more reason to ensure that auto-critique isn't all we do. In Britain, the coalition government is mounting an all-out economic assault on the lives of the most vulnerable of women. From the rise in food banks, to the ways in which women seeking asylum are treated in detention centres such as Yarl's Wood, to the attack on disability living allowance, we seem unable to sustain the same levels of outrage as that directed at us on platforms such as Twitter. Our biggest enemies are the social structures that continue to perpetuate gender inequality, that literally value female lives less. Debates that rage across the virtual world are necessary but cannot operate in a bubble, removed from the lives of the most voiceless of women.

The power of social media remains that it is open to anyone with internet access, free to write what they want and see it resonate with others far and wide. Quelling dissenting voices, offering instead a hollow image of unity at all costs, will never work. Sexist and racist trolls, it seems, know as well as anyone that the very attempt to impose unity often brings the most promising of movements to the brink of disintegration.

But we haven't disintegrated yet. No amount of fabricated fight-mongering by people who don't really care about the issues can ever be as stressful as the real feminist Twitter wars – and no amount of arguing among ourselves is going to stop us. The reason sexist trolls fretting alone in their bedrooms are frightened of political women online, particularly women of colour, is the same reason they won't win. Despite our differences, and even because of our differences, we are powerful, and we are many, and this is our time, not theirs.