Santos is operating the well, in WA-191, about 15km east of its Mutineer oil production facilities.
Tap acquired its 8.2% stake in the WA-191-P permit from Woodside in March, which it said would enhance the commercialisation potential of its Amulet oil discovery in the neighbouring permit.
Fletcher-1 is targeting 20 million barrels of recoverable oil in a large structural closure in Cretaceous aged sandstones.
Drilling is expected to take 13 days to reach 3000 metres.
The WA-191-P JV comprises Santos (operator) and Kufpec, both with 33.4% stakes, Nippon Oil Exploration, with 25%, and now Tap Oil with 8.2%.