Inside the laptop on which I'm writing this column are tiny quantities of tin used to solder parts on the circuit board. I've used my mobile roughly 20 times in the course of my research, calling the US, the Democratic Republic of Congo and researchers in London; I know it's the small element of another crucial mineral, tantalum, that ensures that my mobile phone doesn't lose its memory when the battery goes dead. Both the tin and the tantalum are contributing their part to making my life easier and my work more effective. The painful paradox is that these minerals help make the lives of thousands of eastern Congolese agonisingly wretched.

The minerals are dug by hand from remote mines, often by forced labour. Conditions are dangerous and accidents frequent. Many mines are directly controlled either by corrupt army commanders or armed rebel groups – the difference between the two is perilously hard to pin down. The ores are carried in sacks on porters' backs for up to 45 kilometres to airstrips; at every stage of the process "taxes" and tolls are extracted. Global Witness in a report published tomorrow calculated that the revenue from the major Congolese mining area at Bisie could be in the region of $30m a year. With so much money at stake, conflict over control of these assets is brutal, and the terrorising of civilian populations through rape and murder has become routine.

That much is clear, so what can be done about it? How can I be sure that if I buy a new mobile phone for Christmas it's not contributing to this hell? (My 10-year-old phone is now deemed vintage, such is the pace of electronic consumer fashion.) Millions of sleek, glossy, elegantly designed laptops and mobile phones will end up as presents under Christmas trees all over the globe in the next few weeks, and how will we know what they have really cost?

What lies between my laptop in London and the mines in the three eastern provinces of Congo is an immensely complex entanglement of economics and politics. Think of how kite strings can get tangled and take hours, even days, to unravel and you have the right metaphorical image. This is globalisation in which supply chains crisscross continents, passing from company to company, and at every stage every player has an interest in obfuscation: either blatantly on the ground in Congo, where huge quantities of this million dollar trade are illegal; or closer to home with the polite refusal to engage, the citing of commercial confidentiality.

The obfuscation is hugely convenient. For nearly 15 years of this civil war too many links in the chain have hidden behind the convenience of ignorance: "We just don't know; we can't be sure." It's the excuse that the electronics industry has used; it's the excuse us consumers use. It's the excuse put forward by the traders – known as comptoirs – in the trading centres of Kivu, by the truck drivers transiting Rwanda and by the big smelting companies in China and Malaysia who supply the world's electronic manufacturers. Nor is it entirely unjustified; Congo accounts for only 6% of the world's tin and between 9% and 18% of its tantalum. The huge smelting operations suck in raw materials from all over the world; every electronic product, aeroplane and car could carry traces of conflict minerals.

This kind of complexity is why no one has tried to launch a boycott. But the interesting thing is that you don't need to run a boycott to get big brand names on the run; everyone knows what's on the table. Nervous brand managers watch the success of Uncut's protests closing down branches of Topshop and the widespread newspaper coverage of Philip Green's tax arrangements and shiver with horror. Campaigners on both sides of the Atlantic are using these scare tactics with brilliant effect.

The Enough campaign in the US publishes tomorrow an index measuring the efforts of the electronic companies to clean up their supply chains. It ranks companies such as HP, Intel, Microsoft, Nokia and Dell top, while Sony Ericsson, IBM and Toshiba lag far behind. The plan is that just naming and shaming will ratchet up the pressure, and in turn these companies will lean on the smelting operations that supply the minerals they use.

The biggest achievement of Enough and Global Witness was to get through Congress last summer the Dodd-Frank act which will require all manufacturers to report due diligence (it must be published and must include an independent audit) on their supply chain for the four conflict minerals found in eastern Congo – coltan which produces tantalum; wolframite (tungsten); cassiterite (tin); and gold). It kicks in for 2011, and manufacturers from jewellers such as Tiffany through to Ford Motor Co and Wal-Mart are expected to report. It's probably one of the biggest efforts ever mounted to clean up a supply chain. While there has been some grumbling, key electronic companies such as Motorola actually helped get the act through, increasingly edgy of that moment when images of mobile phones dripping with blood could be plastered over the internet. Any time there is a hint of backsliding, all it takes is a carefully placed article by a celebrity lamenting how their favourite gadget is making them feel guilty, and the point is made.

As big brands lean on their suppliers, the pressure filters through to everyone with a stake in this vast mineral rich area of Africa. Countries in the region are nervous that the growing pressure could lead to a boycott of all Congolese minerals, cutting off a crucial source of income. On Wednesday, several heads of state are meeting at a summit in Lusaka to tackle the issue; meanwhile Canadian and German non-governmental organisations are working with the Congolese government on different aspects of a certification scheme. Last August, the Congolese president, Joseph Kabila, announced a ban on all artisanal mining in eastern Congo as the first step to a clear up – although it was largely ignored by the army.

And this is where all the plethora of schemes now being launched fall short – on the ground in Congo where the state is too weak to fulfil basic functions such as control its army, combat armed groups or effectively monitor exports, as the UN's recent report showed. And the state is weak because it cannot counteract the power structures that benefit from the immense wealth of this minerals trade – put at well over a billion dollars in 2009. One expert in Congo believes that the best way forward now is a dogged, slow process of strengthening the Congolese state, reforming the army and the justice system; it is a risky, unglamorous task that aid agencies shy away from.

Eastern Congo's hell is an instance of how globalisation generates ungovernable spaces. Where there is a collision of desperate poverty, plentiful guns and a world greedy for natural resources, a brutal chaos results. To combat that, it takes a very tenacious sort of global campaigning – bringing to attention each element of the system and the part it can play in leveraging change – and mercifully, that is what is now finally starting to happen.