Five years ago I found my father via Facebook. I had always known he lived in a small African country called Swaziland, but had never seen or spoken to him.

Sadly my father passed away six weeks before I first stepped foot on Swazi soil, but I have visited the country four times since, learning something of this verdant, undulant, landlocked state.

Swaziland is one of the world's last absolute monarchies. The 45-year-old king, Mswati III, enjoys total power over his million subjects. Despite his personal wealth of £65m, two private jets and 15 wives, he rules over one of the planet's poorest nations. Almost 70% of Swazis live below the poverty line, and the country has one of the world's highest HIV infection rates (26%) and the one of the lowest life expectancies (49 years).

Placed strategically between regional powerhouse South Africa and potent Mozambique, resource-rich Swaziland doesn't have to be poor. Coca-Cola has its largest African manufacturing plant here, and activists claim the drinks giant generates a massive 40% of Swazi GDP. However, the monarch's tight grip on power has led to cronyism and an ineffective government.

There have been demonstrations by the usually respectful Swazis in recent years, but protests and strikes are not part of the country's cultural makeup, and are easily shut down by the police. Campaigners have faced arrest and threats of torture.

Under the king's "tinkhundla" system of government political parties are banned from involvement in elections, which are little more than cosmetic anyway. Despite elections in September, his grip on parliament remains tight. He handpicked the prime minister, 10 MPs and most of the cabinet. A 2.4bn rand (£146m) bailout from South Africa remains unpaid because the king refuses to accept mild democratic reforms attached to the loan – after all, he isn't suffering.

If the king is able to ignore his people, his parliament and his rich neighbours, what is the peaceful route to change for Swaziland? When global giants speak, even kings listen – and in the realm of the multinational, few stand as tall as Coca-Cola.

Sold in more than 200 countries, Coca-Cola has an annual turnover of £21bn, more than nine times Swaziland's yearly output (£2.33bn). When accused of propping up the Swazi regime, Coca-Cola simply said it does not get involved with the internal politics of the countries it works in.

I'm afraid this is just a convenient fudge. In 1987 Coca-Cola moved to Swaziland from apartheid South Africa once public opinion made its presence there unacceptable. While many global firms, like Coke, have genuinely good intentions towards the people they employ and the countries they work from, they also exist to suck up resources and turn them into cash. Where these motives collide there can be only one winner.

Coca-Cola's bosses don't have to woo electorates, but protecting the company's brand as "the world's favourite drink" is incredibly important. Between 2002 and 2010 Coca-Cola donated £170m through its foundation to "develop and maintain vibrant, sustainable and local communities". In 2001 the company launched its African Foundation, based in Swaziland, which is the centre of the company's development work in the continent. But if such corporate social responsibility projects are to be more than just cynical PR exercises, corporations can no longer turn a blind eye to abuses in the countries they work in.

A concerted campaign has put Coke and other sponsors of next year's winter Olympics in Sochi under pressure to speak out against Russia's draconian anti-gay laws. A similar campaign to encourage Coke to press Mswati III to make changes could have even more impact.

Most Swazis are still very loyal to the king, who they see as a unifying force. He claimed recently to have been told by God to turn Swaziland into a democratic monarchy. Coca-Cola is in a position to have a quiet word in the king's ear to encourage him, without losing face, to start down the path of democratic change.

Last week Coca-Cola declared it would stop working with subsidiaries involved in land grabs, where land is taken from poor people without consent. This move came after 250,000 people signed a petition led by Oxfam. If thousands of people tweet @cocacola or support campaigns like Sum of Us, Coca-Cola may just listen.

Even an absolute monarch won't be able to ignore a company that delivers such a huge part of his wealth.