The companies, at least initially, had significant shareholders - ENRC was controlled by its trio of founding shareholders and the Kazakh government, while Bumi was effectively controlled by its co-founders the Bakrie family.
ENRC is now set to delist from London after a buyout by the founders and the Kazakh government.
Many companies with majority shareholders already have relationship agreements, but shareholders had often complained that they are either ignored or ineffective.
Bumi, for example, did have a relationship agreement with the Bakrie family, but that deal allowed the family to nominate the chairman, chief executive and chief financial officer.
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The FCA said it would set minimum requirements for relationship agreements, which would for example ensure that minority shareholders could have the ability to veto transactions between a business and its main owner.
"That provides the minority shareholders with some teeth to be able to discipline the controlling shareholder where they breach the relationship," said David Lawton, director of markets at the FCA, which can publicly censure or fine a company which breaks the listing rules.
The Association of British Insurers (ABI), whose members manage nearly $3 trillion of assets, is among those who have called for more protection for minority investors.
"These are all good steps in the right direction, although we would like to see some of this go a little further," said Robert Hingley, director of investment at the ABI, which had suggested the FCA needed the power to punish the controlling shareholder rather than the company.
The rules, first proposed by FCA predecessor the Financial Services Authority, will apply to companies with a "premium" listing, who already face higher requirements on regulation and corporate governance than those with a standard listing.
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Some warned the tougher controls could risk driving businesses away. "It remains to be seen whether, by singling out one type of company ownership for extra rules, the FCA will discourage these companies from listing in London," said Roger Barker, Director of Corporate Governance and Professional Standards at business group the Institute of Directors.
Under the changes, minority shareholders will also have greater power over the election of independent directors, who will have to be approved by both the shareholders as a whole and the minority group.
The FCA stopped short of increasing the minimum "free float", the proportion of a company's shares which must be freely available to trade, from its current level of 25 percent to the as much as 70 percent requested by some investors.
It said this seemed too blunt a tool, and that it needed to balance protections with unnecessarily increasing the regulatory burden on companies that are already well governed.
Shares subject to a lock-up of more than 180 days will not count towards the free float level, however, the regulator said, and it is also further consulting on criteria it should take into account when considering waivers in individual cases.
There are around 50 premium-listed companies on the LSE which have a controlling shareholder. Those who do not already meet the new rules will have six months from the mid-2014 implementation date to ensure they comply.