About 250 workers, including pipe fitters, welders and scaffolders have walked out at the Air Products' factory, in Billingham, near Stockton.
The dispute concerns bonus payments after work was affected by bad weather.
Bosses say they believe the strike has not been sanctioned by unions and expect workers to return next week.
A company statement said: “Foster Wheeler, the main engineering procurement and construction management contractor at Air Products' factory, has confirmed that certain construction contractors have taken unofficial industrial action.
“Air Products, Foster Wheeler and the construction contractor management have held a series of talks with unions in recent weeks and all parties are pressing for an early resolution.
“Work continues unaffected throughout the rest of the site.”
The advanced gasification plant is expected to start operations later this year, and, when finished, will be one of the largest of its kind in the world.
Bosses say it will take care of up to 350,000 tonnes of local waste every year, delivering enough electricity to power about 50,000 homes.