The crew is sailing in a 25ft (7.6m) open-deck boat with two small sails. They are hoping to sail from Tonga to West Timor in seven weeks.
The expedition has tried hard to recreate the conditions which confronted the Royal Navy captain.
Memorialised in two Hollywood films as well as novels and poems, the mutiny on the Bounty unfolded in 1789 when William Bligh was cast adrift in the South Pacific by his rebellious crew.
Accompanied by 18 men, Bligh managed to sail from near Tonga to West Timor in a voyage that lasted almost 50 days.
He survived by catching fish and drinking rain water.