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“No one has suggested that Hillary Clinton knew of Thompson’s illegal activities,” said Chuck Thies, manager for the Vince Gray 2014 campaign. “Mayor Gray has not been afforded the same presumption of innocence.”

The Center for Public Integrity reported last year that Thompson and other donors who listed his accounting firm as their employer gave at least $514,350 to federal candidates and political action committees since the 2002 campaign cycle. The biggest beneficiary of those funds was Mrs. Clinton, whose campaigns took in $50,400. Her presidential campaign committee received $40,300 from employees at Thompson’s firm in November 2007, when she attended a fundraiser at the company’s headquarters in the District.

Other recipients included Mr. Obama ($14,500 in the 2008 cycle), Democratic presidential candidate John F. Kerry ($20,000 in the 2004 cycle), presidential candidate John McCain ($13,800 in the 2008 cycle) and Sen. Harry Reid ($9,600 in 2009).

Thompson personally made political contributions totaling at least $215,000 in that period.

Craig Holman, a campaign finance specialist at the watchdog group Public Citizen in Washington, said shadow campaigns are becoming common with the expansion of political action committees.

“Of all the super PACs involved in the 2012 federal elections, 58 percent supported a single candidate — established and run by friends, family or former campaign staff of that single candidate. This is a shadow campaign,” he said.

Mr. Holman said courts have imposed a high barrier in defining which shadow campaigns are legal and which are not.

“As long as there is no traceable agreement between the candidate and the political operative — just a wink and a nod — the courts are reluctant to find coordination between the two,” he said.

Mr. Holman said the public should raise concerns about shadow campaigns, especially those backing every major recent presidential campaign, including Mrs. Clinton‘s.

“Hopefully, public disgust will grow against the artificial line drawn in the sand by the courts as to ‘legal’ versus ‘illegal’ shadow campaigns,” he said.