JEFFERSON CITY, Mo. (AP) — On the heels of a U.S. Supreme Court ruling weakening public-sector unions, labor's clout is being put to a new test by a voter referendum in Missouri over whether the state should ban compulsory union fees in all private-sector workplaces.

The statewide vote in Tuesday's primary on a so-called right-to-work law could be a watershed moment for unions, if they can halt what has been a steady erosion of strength in states with historically deep-rooted support.

"The timing of this is essential. I think everyone wants to write the labor movement's obituary," national AFL...