Say, let each state send representatives to bid for import quota of specific categories of merchandises, therefore making intra-US cooperation among states into inter-state competition; just like how countries have trade deals with EU members while maintaining a relationship with EU.
Under Article I, section 10 of the US Constitution, no state can enter into an agreement with a foreign power without the approval of Congress. No state can enter into a treaty with a foreign power period, even with Congressional approval. This is specifically to prevent what you're talking about: the US is a single country and is supposed to act as a single country on the international stage.
Legally binding agreements between the states are called Interstate Compacts. Creating them requires an act of Congress (Article 1, Section 10). Some are based on a model agreement, approved by Congress, that any state may join, such as the Driver License Agreement. Prior to the original Driver License Compact, it was not unknown for out-of-state drivers to be arrested for minor traffic offenses - because there was no way to ensure that the traffic violator could be held accountable when they left the state. The remainder have to be submitted to Congress for approval and ratification (for example, the New York Port Authority was approved by Congress in 1921). The only compact that I am aware of, that includes parts of another nation, is the Great Lakes Charter, which created the Great Lakes Compact governing water rights and water usage in and around the Great Lakes bordering USA and Canada.