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Archbishop Justin Welby: The 'muscular Christian’
"Investigations are ongoing to find out details of the blasts and the motive behind them," assistant police commissioner Mkadam Khamis told reporters.
One blast took place at the Anglican cathedral, a historic building in the heart of the narrow and winding ancient streets of Stone Town, the UNESCO-listed historical centre of the capital of the semi-autonomous Tanzanian archipelago.
The other occurred at the seafront Mercury's restaurant and bar, a favourite of tourists.
"It is suspected that the explosives are homemade bombs thrown by unidentified thugs," Khamis said, adding that the blast at the cathedral had caused only minor damage to parked cars.
The bar, located near the main port and seafront square, is named after flamboyant rock musician and Queen frontman Freddie Mercury, who was born on the island.
The British High Commission said it was "ready to provide consular assistance as necessary".
Last year unknown attackers hurled acid into the faces of two British teenage girls as they strolled through Stone Town, as Zanzibar's Muslim majority were preparing to celebrate the end of the Islamic holy month of Ramadan.
In the past year, attackers have also thrown acid into the faces of religious leaders, both Christian and Muslim. Others have been shot.