Robert E. Scott, a senior economist at the left-leaning Economic Policy Institute, estimates that NAFTA is responsible for the net loss of roughly 700,000 American jobs to Mexico, while China’s admittance to the W.T.O. cost the United States more than three million jobs. Roughly three-quarters of those losses were in the more heavily unionized manufacturing sector, contributing to the steep decline in private sector union membership, which went from 15.7 percent in 1993 to a nadir of 6.6 percent in 2014, the lowest figure in a century.

Mr. Scott’s research has shown that these agreements not only drive down manufacturing wages, but they also have a ripple effect, pushing down the pay in other jobs typically held by workers without a college education: home health care worker, truck driver, waitress. A 2013 paper by Mr. Scott’s colleague at E.P.I., Josh Bivens, found that, on average, noncollege-educated American workers, the people who make up roughly 70 percent of the labor force, lose nearly $2,000 a year in wages owing to the growth of trade with low-wage countries promoted by free-trade agreements.

The depth of Mrs. Clinton’s estrangement from labor may not be known until April 5, when Wisconsin holds its primary. Since 1960, no Democrat has won the general election without winning the state, and a loss to Mr. Sanders in Wisconsin could foreshadow trouble against Donald Trump, whose opposition to free trade helped propel him to victory in Michigan. Exit polls there showed that a majority of Republican voters also believe that free trade takes away American jobs. Mr. Trump decisively won that group. “You know, Michigan has been stripped,” Mr. Trump told CNN’s Anderson Cooper the day after his victory. “You look at those empty factories all over the place. And nobody hits that message better than me.”

While Mrs. Clinton’s pro-union shout out at the debate resonated widely, many of Wisconsin’s labor activists remain skeptical. “A lot of our job problems stem from NAFTA, and the TPP will kill us,” Gerry Miller, a United Steelworkers welder at a Caterpillar plant in South Milwaukee, told me last month. “We can’t compete with people being paid two dollars a day in Vietnam. The thing that we’re most upset about is the pandering. Democrats like Clinton speak labor out of one side of their mouth, but the corporate interests pull the strings.” (Mr. Scott estimates that adoption of TPP will result in the net loss of roughly 40,000 jobs in Wisconsin, 215,000 in Michigan and 113,000 in Ohio.)

While Mrs. Clinton has received the endorsement of many of the large national unions, the A.F.L.-C.I.O. has not yet taken sides. Many union locals have chosen to back Mr. Sanders. David Poklinkoski, the president of IBEW Local 2304, a Wisconsin utility union, said his local had never endorsed anyone for any office before, but recently passed a unanimous resolution endorsing Mr. Sanders. Mr. Poklinkoski praised the senator’s consistent opposition to free-trade agreements.

After the Milwaukee debate, Mr. Poklinkoski told me that two of his members who watched it came away as Sanders supporters. But Mr. Poklinkoski was alarmed to hear that the men’s second choice was Mr. Trump. Mr. Poklinkoski believes Mrs. Clinton could be vulnerable in Wisconsin.

“I’m worried about Trump versus Hillary,” Mr. Poklinkoski said. He noted that at home Governor Walker had successfully portrayed himself as an anti-tax, blue-collar politician, an image that helped him divide Wisconsin’s workers during the state’s labor battles. “If you have a right-wing populist, you can beat a corporate Democrat,” Mr. Poklinkoski said. “Scott Walker did it three times here.”