A spokesperson for General Electric says that the company is a totally different one than it was in 2005, after several divestments of U.S. assets, like NBC in 2009, which did not involve job losses, reduced its U.S. employee count. It also has purchased businesses abroad like Alstom which has increased its foreign headcount. Honeywell also points the finger at divestitures of domestic businesses, transactions that don't necessarily result in job losses, as the reason for it's falling U.S. headcount. Lear—the company on our list with the highest share of overall foreign employment—said said the automotive industry markets it supplies that are outside the U.S. are growing faster than the domestic market. Mondelez didn't respond to requests for comment.

Even in the cases of these firms, where we have hard, public data, the issue of acquisitions and divestitures make identifying which, if any, major companies are "shipping jobs overseas" a difficult, if not impossible one to answer. Here's what else we found:

A severe lack of information: fewer than one-fifth of companies studied voluntarily reveal where their employees are located. Because there is no requirement for disclosing where public companies employ workers, it's impossible to study the problem in a systematic way, or know which companies are the worst offenders. Labor leaders have been lobbying to make the disclosure of outsourcing mandatory. The president may consider taking up this banner.

Car companies are being unfairly singled out: Take GM for example. The company has reduced U.S. employment over the past ten years, but it also cut jobs abroad too. Foreign jobs are shrinking at a slower rate, sure, but much of the story of disappearing manufacturing jobs is about automation rather than outsourcing. If car companies are reducing the number of workers overall that they need to produce ever more cars, it will not be very fruitful to focus as much energy on the industry as the Trump Administration has.

Motivation is paramount: When one thinks of outsourcing the typical picture is of employers who shutter a factory in the U.S., move it abroad, and then import those products for sale in America. This is exactly the type of behavior the president says he is targeting. But some international trade experts argue that these sort of moves are less common today than ten or twenty years ago. Harry Moser, a former manufacturing executive and president of the reshoring initiative says that the U.S. has, since 2010, seen more jobs come back to the U.S. than leave on the strength of cheap energy and the ability to automate tasks that once had to be done by hand.

Meanwhile, the majority of the foreign hiring in the financial statements reviewed by Axios appears to be motivated by a different concern: the desire to sell to foreigners, not Americans. Because developing economies grow at a faster rate than developed economies like the U.S., returns on investment will be higher abroad than at home. That means companies will naturally increase employment abroad, but those jobs were never going to go to Americans anyway. Starbucks illustrates this point, as it has increased the number of foreign employees by 278% since 2005, but it's not as if a displaced worker in Michigan can take a job serving coffee in Bangalore.

Why it matters: Trump has gone all-in on the idea that he'll bring American manufacturing back by sheer force of will. But much of these jobs left decades ago, and are now being done by machines. The sort of expansion Corporate America is doing abroad isn't the sort that Trump can easily vilify, and those companies that are engaging in this behavior aren't required to disclose it.