The adversaries in the shareholder dispute include U.S. venture capitalist Paul Tierney, a former Peace Corps volunteer and former owner of D.C. United soccer team. Along with his son Matthew, he accuses the man behind EcoCash, Strive Masiyiwa, a jet-setting philanthropist who sits on the board of Unilever and the Rockefeller Foundation, of trying to use an unusual debt-for-equity swap to grab a bigger piece of EcoCash now that it has become a key piece of monetary infrastructure.

Mr. Masiyiwa is the founder and biggest shareholder of EcoCash’s owner, Zimbabwean cellphone operator Econet Wireless Zimbabwe Ltd. The company’s shares, which trade on the Zimbabwe Stock Exchange in dollars, have increased their value nearly 10-fold in the past 18 months and were recently worth more than $7 billion on paper.

Representatives for Mr. Masiyiwa say the disgruntled shareholders are failing to understand the complexities of operating in one of the world’s most distorted economies.

After the collapse of the Zimbabwe dollar in 2009, the country adopted the U.S. dollar to stabilize prices. But since it can’t print them itself, the country is now so low on dollars that banks have stopped dispensing cash or transferring money abroad.

The government, under budgetary pressure, has essentially created quasi-U.S. dollars that exist only electronically in local bank accounts and are now coursing through the economy. But they are deeply devalued: On the black market, a $100 bill can cost as much as $350 in digital dollars.

Launched in 2011, EcoCash allows users to trade electronic dollars over their phones. Users set up a mobile wallet—either linked to a bank account or topped up through transfers from another user—and then use the service to make everyday payments by punching codes into the phone’s call function.

Eight out of 10 transactions in the Southern African nation—whether paying for milk at the corner store, paying an electricity bill or buying fuel on the black market—are done using EcoCash, according to data from the Reserve Bank of Zimbabwe. In a country of 13 million people, the platform has more than eight million users.

“This is a very, very unique story in the world,” said Roy Chimanikire, Econet’s finance director. “This is an asset which basically is running the financial system of the country, that has stopped the country from collapse.”

The Zimbabwe government has encouraged electronic payments to help deal with the cash crisis, but is trying to diversify beyond EcoCash, including by promoting the use of regular credit and debit cards.

EcoCash’s move into the heart of Zimbabwe’s financial and monetary system attracted international investors, among them the Tierneys, betting on an economic recovery following the ouster of longtime strongman Robert Mugabe in 2017.

But the currency crunch behind EcoCash’s meteoric rise is also at the root of the shareholder battle now gripping its parent company.

In February 2017, with Zimbabwe’s version of the dollar dropping in value and the company struggling to pay back foreign loans, Econet raised $130 million through a rights issue, in a mix of local quasidollars and real U.S. dollars. Investors received shares for 5 cents and debentures, a debtlike instrument with an issue price of 4.7 cents. The debentures were meant to be paid back in six years. Investors soon deemed the debentures worthless after the government blocked a plan to let Econet back them with dollars held abroad.

Mr. Masiyiwa ended up holding 56% of all newly issued shares and debentures.

EcoCash’s usage has soared since then and so has Econet’s share price, trading as high as $2.85 in October.

In November, the company proposed converting the debentures into equity. Minority investors, some of whom bought shares after the rights issue, and thus didn’t own the debentures, cried foul.

“There is no justification for this blatant disregard for the rights of minority shareholders,” Paul and Matthew Tierney wrote in a Nov. 27 letter to the Zimbabwe Stock Exchange.

They said Mr. Masiyiwa, already worth $1.7 billion according to Forbes, was using those debentures to grab a bigger stake in the company at a steep discount. Mr. Masiyiwa would turn the debentures he purchased for just $28 million in 2017 into shares valued around $1.3 billion on the day the conversion was proposed.

If the deal went through, it would lift Mr. Masiyiwa’s stake in Econet to 46.7%, from 42.6%, while diluting the holdings of investors who don’t own any debentures.

Other investors unhappy with the proposed conversion include Zimbabwean financial-services companies Old Mutual Zimbabwe and Imara Securities as well as South Africa’sCoronation Fund Managers Ltd. , Sanlam Ltd.and Allan Grey, which together own some 25% of Econet.

Mr. Masiyiwa “treats Econet as his personal company and minority investors get stiffed,” said Peter Townshend, portfolio manager for Sanlam’s Africa Equity Fund.

Amid the investor outcry, a vote on the proposal has been delayed twice, as recently as Dec. 14. Econet says it is still considering the debenture swap but hasn’t set a date for a fresh shareholder vote.

Econet and representatives of Mr. Masiyiwa’s holding company say the proposed conversion would benefit a big majority of Econet’s shareholders and reward investors who helped the company during tough times.

Complicating matters, in December, Econet spun off the EcoCash business, along with some smaller financial-services ventures, into a separate company listed on the Zimbabwe Stock Exchange. Dubbed Cassava Smartech, it immediately became the country’s largest traded company with a market capitalization of $3.87 billion.

These gains may just be mostly on paper anyway. Zimbabwe investors have piled into assets denominated in dollars, including stocks, that they think may retain value in case the country decides to depeg from the U.S. dollar.

“We have a U.S. dollar operating in Zimbabwe, but actually that U.S. dollar is not convertible into a U.S. dollar outside Zimbabwe,” said Mr. Chimanikire, Econet’s finance director. “So it’s not really a U.S. dollar.”