Some union members want federal labor regulators to make the International Association of Machinists and Aerospace Workers (IAM) hold the vote again.
They say thousands of workers were effectively disenfranchised because they were out of town when the vote was held.
About 23,900 of the more than 32,000 members of District 751 of the IAM cast votes. That is a few thousand votes fewer than in November, when members rejected Boeing's earlier offer.
Four union members have filed charges with the National Labor Relations Board, alleging that the IAM's international leadership didn't fairly represent its members by forcing the vote, which local union leaders opposed.
More than 30 Machinists rallied Monday outside their union hall in Everett, calling for a new vote on Boeing's offer.
"Unions have a lot of leeway when it comes to representing their members," said Charlotte Garden, a law professor at Seattle University's School of Law. "The board will be looking to see if the union acted arbitrarily or in bad-faith."
Dan Catchpole: 425-339-3454; email@example.com.
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