Initially, the coming Zero Government Society will relate to foreigners that way.
No domestic government will exist to interfere with the passage of goods, services or people across the border of the Former USA with neighboring, government-infested societies, so freedom of trade will prevail to the extent that those other governments allow it. There will be no US tariffs, prohibitions, inspections, licences whatever. If a buyer and a seller agree on terms for a transaction across that border, it will proceed; as it should, since they are the only two parties properly involved. Only the foreign government(s) will continue to be a nuisance. Liberty will enlighten the world.
The result will be a useful boost to living standards here and abroad, for trade will increase as restrictions on it disappear; we shall be better able to buy from the most efficient producers and sell to buyers who most want goods we make. This highly sensible division and specialization of labor will reach its full potential when foreign governments also vanish, but some benefit will seen at once.
I don't think it will happen, but this improvement could be obtained today, if the FedGov were to dismantle its "Customs" apparatus and allow trade to take place as easily across the National border as it does across State lines domestically. It's unlikely because governments never volunteer to give up power; they get their kicks from exercising it, by saying "Yes" to some and "No" to others. But the recent G-7 meeting in Canada reminded me that such a thing is not impossible.
President Trump is not happy with the various inter-government trade agreements his predecessors have made, and is busy trying to scrap and renegotiate them. That makes him a pariah in meetings like that G-7. He reciprocated by insulting it by arriving a little late; the expression on Christine Lagarde's face must on its own have made his trip worthwhile. Lagarde is the long-time Managing Director of the IMF, a group that coordinates the theft of large sums of money from taxpayers in the most productive nations and gives it to politicians in less productive ones.
Trump may be right in his dislike of the terms of those agreements, but his apparent solution - to renegotiate them - is not the fix that's needed. If he can overcome the massive prejudice against him in D.C., the adequate and simple solution would be for the FedGov to declare that the USA is a Free Trade area and scrap all restrictions on international trade. It's interesting to try to predict what would then take place.
With all import tariffs and impediments removed, the first result would be that US customers will find it easier to buy goods from overseas; their prices will fall. So the volume of imports will increase.
That will in turn create a demand for more foreign currencies, to pay for the increase. US buyers (via bankers, probably) will need to obtain Euros, Yuan, Yen and Pounds, to pay for those imports in a manner acceptable to the various vendors; a US "dollar" is not useful in Denmark. Now, when demand for something rises and its supply is fixed or nearly so, its price rises. Therefore, the cost of Euros and Yen etc in terms of the dollar will increase.
Accordingly, other factors being static, the price of foreign goods in dollar terms will stop falling. An equilibrium will be reached. The rate of exchange among currencies will have changed, but so what?
In due course, it's likely that residents of foreign countries will note the increase in American standards of living that followed the abolition of import tariffs and controls, and demand the same of their own governments, for as long as they remain in place. When those too have vanished, the exchange rates will adjust again, in reverse, and standards of living will rise there too, rounding out a worldwide increase in prosperity. The only losers will be governments, whose revenues from cross-border taxes will vanish. Sob, sob.
Unilaterally scrapping these controls is not just good economic theory. One the most beneficial law-endings of the last thousand years was the British repeal of the Corn Laws in 1846. Those laws had imposed tariffs on imported grain, so "protecting" domestic growers from low-price imports. When scrapped, food became significantly cheaper in Britain and the same principle (free trade) was applied to other products without reference to what other governments did. The result was greatly to magnify the increase in living standards brought by the Industrial Revolution - an increase that, for the first time ever, coincided with dramatic rises in population and life expectancy.
It so happens that that country has a comparable opportunity right now, to withdraw from the stifling rules of the European Union (as demanded in the 2016 referendum) without any protracted negotiation of a trade deal; its government could simply declare the withdrawal done and dusted, and publish a one-page declaration that no restrictions or taxes will impede any imported goods or services to the United Kingdom. As Milton Friedman proposed in his Capitalism and Freedom (ch.4), they "could say to the rest of the world: We believe in freedom and intend to practice it."
Whether or not they will, is unknown; but come the ZGS the decision will be removed from government hands. When they evaporate, starting I think in the US, all their destructive restrictions will go with them.