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The country's main trade-union confederation, Cosatu, which represents 1.7m workers, hints that it may join in.
ECONOMIST: A health tragedy compounded by government farce
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"We are just too tired of getting blackmailed all the time, " Zwelinzima Vavi, the General Secretary of COSATU, told the BBC's World Business Report.
BBC: South African unions claim strike success
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The above-inflation pay increases that Cosatu has claimed over the past five years have swelled government expenditure and crowded out spending on worthier causes.
ECONOMIST: South Africa's politics
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In the past five years the Congress of South African Trade Unions (Cosatu) has been criticised by many of its members for not fighting their corner hard enough.
ECONOMIST: South Africa
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South Africa's trade union confederation, Cosatu, also expressed concern that Mr Tutu "has allowed his passionate personal hatred of the ANC and its President Jacob Zuma to cloud his judgement".
BBC: ANC condemns Tutu's 'sacrilege'
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"When a very similar incident occurred in India recently, there was a massive outbreak of protest and mass demonstrations in the streets, " Cosatu spokesman Patrick Craven said, AFP news agency reports.
BBC: Jacob Zuma (file photo)
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Some leading lights in Cosatu do not support him.
ECONOMIST: South Africa: It��s not just the mines | The
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The Congress of South African Trade Unions (Cosatu) demanded an excessive 8.6% pay rise, more than double the rate of inflation, at a time when the country's recovery from last year's recession is fragile.
ECONOMIST: South Africa's politics
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On Wednesday, the governing African National Congress (ANC) party and the Congress of South African Trade Unions (Cosatu) demanded an investigation into how the guests of the family flew from India to the Waterkloof Air Force Base near Pretoria.
BBC: South Africa probes India over Gupta wedding