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...