What might happen is that different parts of the United States would have different foreign policies. This would cause a great deal of confusion as well as problems for the United States.
For example, early in the history of the US, different states would probably have taken different sides in the conflict between France and Britain. New England was very pro-British and might have taken their side while other states sided with France.
In today's world, this would be less likely. However, one can imagine a scenario where some states would enter into free trade agreements with other countries. This would then cause real problems because products could enter those states and then be shipped from there to other states that did not want free trade.
It is for reasons like these that the Constitution prohibits the states from making treaties.