From a townhouse near a megachurch in Atlanta, Kelvin Lyles recruited about 300 accomplices to embark on a crime spree. His group scammed ATMs, internet retailers and credit-card companies, grabbing around $350,000, until late 2015, when federal agents closed in.

Mr. Lyles was the only one convicted. None of his accomplices existed.

In a twist on ID theft, criminals are deploying figments of their imaginations, in what is often called synthetic-identity fraud. It’s one of the fastest growing forms of identity crimes, the Justice Department says, and among the hardest to combat.

Because the person taking out cards or loans isn’t real, there are no consumer victims to alert lenders. When companies and law enforcement discover something amiss, they often wind up chasing ghosts. Mr. Lyles secured credit cards often using fictional names and numbers the Social Security Administration hadn’t yet assigned.

Synthetic-identity fraud exploits a vulnerability in America’s consumer-credit system. Lenders often consider a loan applicant legitimate if the applicant has a credit report at Equifax Inc., TransUnion or Experian PLC. But a new “credit file”—essentially a precursor to a credit report—often gets created when someone simply applies, even if the loan doesn’t come through.

Some lenders approve loans after reviewing credit files, which helps turn those files into full credit reports. That’s how a fictitious person, or 300 fictional people, can end up with a credit card.

While a small part of total identity-fraud losses—that number hit about $16.8 billion in 2017, according to consulting firm Javelin Strategy & Research—synthetic-identity losses are soaring.

TransUnion says a record $355 million in outstanding credit-card balances was owed by people who it suspects didn’t exist in 2017, up more than eightfold from 2012. It estimates lenders have issued credit cards or loans to millions of synthetic identities in the U.S.

In January, Accenture PLC listed synthetic-identity fraud as one of the biggest threats facing banks in 2018, saying it would be “costing banks billions of dollars and countless hours as they chase down people who don’t even exist.”