Democratic National Committee Chairman Tom Perez on Thursday called for a review of the individual precinct results from the botched Iowa caucuses, adding fresh uncertainty to a process already plagued by chaos.

The state party, though, said it would take a request from one of the candidates’ campaigns for a “recanvass” of the results — a step the 2020 Democratic contenders weren’t necessarily eager to take.

“Enough is enough,” said Mr. Perez, who has been a target for criticism since the caucuses went off the rails Monday, though it is a state party operation.

He demanded the Iowa Democratic Party do the precinct-level review to help “assure public confidence in the results.”

The recanvass would entail reviewing the worksheets from each of the approximately 1,700 caucus sites, a process some local party officials were seemingly already going through.

With 100% of precincts reporting, Sen. Bernard Sanders of Vermont held a slight advantage over former South Bend, Indiana, Mayor Pete Buttigieg in what Mr. Sanders called the “popular vote.”

But Mr. Buttigieg held a 0.1% lead over Mr. Sanders in projected “state delegate equivalents,” which are tied to the number of delegates to the Democratic National Convention that will be allocated to the candidates.

With Mr. Sanders and Mr. Buttigieg both claiming victory, it appears unlikely that either one of them would want to request a recount and risk a material change in the outcome.

Asked about Mr. Perez’s call for a recount, Mr. Sanders said he doubted it would change the outcome.

“At the end of the day, what will, I expect, almost definitely happen is that Mr. Buttigieg and I will end up with the same amount of delegates,” he said. “What certainly is not going to change is the fact that in terms of the popular vote we won a decisive victory.”

Sen. Amy Klobuchar of Minnesota, who was running in fifth place in Iowa, said that “you have to make sure that every single vote was counted.”

“Sometimes in caucuses things can be close,” Ms. Klobuchar told NBC News. “You have to go back in and figure it out.”

Troy Price, chairman of the Iowa Democratic Party, said they’re prepared to address such a request from one of the presidential campaigns. The party then would have 48 hours to respond to a “valid, written” request for a recanvass signed by a candidate.

“In such a circumstance, the IDP will audit the paper records of the report, as provided by the precinct chairs and signed by representatives of presidential campaigns,” Mr. Price said. “This is the official record of the Iowa Democratic caucus, and we are committed to ensuring the results accurately reflect the preference of Iowans.”

The election night disaster has been an embarrassment for Iowa Democrats and a low point for Mr. Perez.

Marianne Williamson, the self-help author who ended her presidential bid last month, said on Thursday that Mr. Perez needs to go.

“The only way for the Democratic party to regain and retain the trust of its own members is for Tom Perez to resign immediately, the DNC to take full responsibility for this mess, and for the American people to see a realignment of the process with fairness and integrity,” Ms. Williamson tweeted.

The difficulties arose in part when local officials had problems using a new app that was intended to streamline the process of reporting results. Local Democratic officials then had trouble relaying the results to the state party via phone as a backup.

Mr. Price said Thursday that the party had found “inconsistencies” in the data and used paper records to “promptly” correct the errors.

The debacle fueled relentless mocking from President Trump and Republicans, who have questioned whether a party overseeing such a mess is competent enough to be trusted with major policy decisions on health care and other issues.

Two top House Republicans sent a letter on Thursday to Gerard Niemira, the CEO of Shadow Inc. — the company behind the app Iowa Democrats used for the caucuses — expressing “grave concern” about the service.

“Unfortunately, by choosing not to test your app for any technical glitches prior to its roll-out, your company has caused significant doubt over the Caucus results and, consequently, has undermined electoral confidence,” wrote GOP Reps. Mike Rogers of Alabama and Rodney Davis of Illinois.

The company said it regretted the delay in the reporting of results, but said the underlying data and collection process was sound.

Still, the Nevada State Democratic Party made clear that it will no longer be using the same app or vendor for its Feb. 22 caucuses.

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