Donald Trump’s claim that illegal voting may have cost him a popular-vote majority has touched off outrage. Widespread voter fraud, the media consensus suggests, isn’t possible. But there is a real chance that significant numbers of noncitizens and others are indeed voting illegally, perhaps enough to make up the margin in some elections.

There’s no way of knowing for sure. The voter-registration process in almost all states runs on the honor system. The Obama administration has done everything it can to keep the status quo in place. The Obama Justice Department has refused to file a single lawsuit to enforce the requirement of the National Voter Registration Act that states maintain the accuracy of their voter-registration lists. This despite a 2012 study from the Pew Center on the States estimating that one out of every eight voter registrations is inaccurate, out-of-date or duplicate. About 2.8 million people are registered in more than one state, according to the study, and 1.8 million registered voters are dead. In most places it’s easy to vote under the names of such people with little risk of detection.

An undercover video released in October by the citizen-journalist group Project Veritas shows a Democratic election commissioner in New York City saying at a party, “I think there is a lot of voter fraud.” A second video shows two Democratic operatives mulling how it would be possible to get away with voter fraud.

The Justice Department has opposed every effort by states—such as Kansas, Arizona, Alabama and Georgia—to verify the citizenship of those registering to vote. This despite evidence that noncitizens are indeed registering and casting ballots. In 2015 one Kansas county began offering voter registration at naturalization ceremonies. Election officials soon discovered about a dozen new Americans who were already registered—and who had voted as noncitizens in multiple elections.

How common is this? If only we knew. Political correctness has squelched probes of noncitizen voting, so most cases are discovered accidentally instead of through a systematic review of election records.

The danger looms large in states such as California, which provides driver’s licenses to noncitizens, including those here illegally, and which also does nothing to verify citizenship during voter registration. In a 1996 House race, then-challenger Loretta Sanchez defeated incumbent Rep. Bob Dornan by under 1,000 votes. An investigation by a House committee found 624 invalid votes by noncitizens, nearly enough to overturn the result.

How big is this problem nationally? One district-court administrator estimated in 2005 that up to 3% of the 30,000 people called for jury duty from voter-registration rolls over a two-year period were not U.S. citizens. A September report from the Public Interest Legal Foundation found more than 1,000 noncitizens who had been removed from the voter rolls in eight Virginia counties. Many of them had cast ballots in previous elections, but none was referred for possible prosecution.

The lack of prosecutions is no surprise. In 2011, the Electoral Board in Fairfax County, Va., sent the Justice Department, under then-Attorney General Eric Holder, information about 278 noncitizens registered to vote in Fairfax County, about half of whom had cast ballots in previous elections. There is no record that the Justice Department did anything.

A 2014 study by three professors at Old Dominion University and George Mason University used extensive survey data to estimate that 6.4% of the nation’s noncitizens voted in 2008 and that 2.2% voted in 2010. This study has been criticized by many academics who claim that voter fraud is vanishingly rare. Yet the Heritage Foundation maintains a list of more than 700 recent convictions for voter fraud.

A postelection survey conducted by Americas Majority Foundation found that 2.1% of noncitizens voted in the Nov. 8 election. In the battleground states of Michigan and Ohio, 2.5% and 2.1%, respectively, of noncitizens reported voting. In 2013, pollster McLaughlin & Associates conducted an extensive survey of Hispanics on immigration issues. Its voter-profile tabulation shows that 13% of noncitizens said they were registered to vote. That matches closely the Old Dominion/George Mason study, in which 15.6% of noncitizens said they were registered.

Fixing this problem is very straightforward. The Trump administration should direct the Department of Homeland Security to cooperate with states that want to verify the citizenship of registered voters. Since this will only flag illegal immigrants who have been detained at some point and legal noncitizens, states should pass laws, similar to the one in Kansas, that require proof of citizenship when registering to vote. The Justice Department, instead of ignoring the issue, should again start prosecuting these cases.

The bottom line is that the honor system doesn’t work. There are people—like those caught voting illegally—who are willing to exploit these weaknesses that damage election integrity.