The deal saves 358 of 509 jobs after owner Toridoll moved to appoint administrators, while nine sites including Spitalfields, Glasgow and Edinburgh are set to close.
Fulham Shore said a pre-pack administration and sale would put the 28-site chain on a more sustainable footing and let it focus on Franco Manca.
The Real Greek, founded in 1999, last reported a £3.6m operating loss as UK hospitality businesses faced rising energy, food, labour and business-rate costs.
With The Real Greek saved but many sites closed, is this the start of a new wave of hospitality takeovers or just delaying more closures?
How might Karali Group’s strategy for The Real Greek differ from previous owners, and can it overcome the sector’s 'perfect storm' of rising costs and shrinking demand?
Could major tax and policy reforms reverse the decline of UK restaurants, or is a fundamental change in consumer habits to blame?