Congress gets involved, and there's a second Nullification Crisis. South Carolina and a few other states go to bat for whichever state was on the losing end of the congressional decision (let's say Michigan), and Darth Jackson loses his cool. Bam. Civil war over states' rights in 1836, but with the added factor that several northern states are involved. Jackson declares effective martial law, and uses the military intervention to start messing with his political opponents (mainly the Whigs, with a few others besides- Jackson made a lot of high-level enemies). However, several Whig military officers (led by Winfield Scott, amongst others) refuse Jackson's orders as clear violations of the Constitution ('unlawful search and seizure' or something) and the United States descends into a three-sided civil war between the Whigs, the Democrats, and the states'-righters.
The European powers look on the affairs with great interest. Mexico uses the whole thing as a distraction and completely destroys the Texicans at the Alamo and elsewhere. The British use the matter to complete the acquisition of the whole of Oregon Country and begin quietly interfering with the northern states, supporting whichever side looked weaker at the time.
Darth Jackson hears of this and decides that he can spare enough men to skirmish with the Canadians. They do so, and Britain is dragged into the war.
British forces, in a repeat of 1812, bombard and burn cities along the coast in Jacksonian hands, including Washington, DC. Jackson surrenders shortly thereafter, but the States' Righters continue the fight against the Whigs and the remnants of the Jacksonians. This fight, though, quickly ends, and Clay, Calhoun, Webster, Quincy Adams, and Scott form a provisional government until such time as they can either amend or rewrite the Constitution to prevent such a terrible tragedy from happening again.
And Ohio retains the strip.