What do you think would have happened had the Anti-Federalists, rather than the Federalists, prevailed in the ratification process of the Constitution? What kind of government would they have shaped? How would that government have dealt with the difficult issues facing the new republic—slavery, concerns about mob rule, and continuing hostility in the international community?

Federalists believed in the establishment of a strong central government, and the US Constitution was devised as a way of achieving this goal. Anti-Federalists, on the other hand, feared that this would enable the kind of tyranny from which Americans had freed themselves in their war against the British. They favored a looser, more decentralized system of government in which ultimate sovereignty remained with the individual states. Had they prevailed, the United States would've been a very different country indeed. That's not to say that the existing constitutional arrangement, the Articles of Confederation, would've been retained in their entirety, but the underlying principles on which they were based certainly would have.
Legal and political sovereignty would still have resided with the states under the anti-Federalists. In relation to slavery, this would have meant no attempt being made by the limited federal government to interfere in slavery and its expansion. The issue of slavery, like all important matters, would be left to the states to decide.
Federalists were anxious to control what they saw as outbreaks of mob rule. Indeed, one such outbreak, Shay's Rebellion, was one of the main factors that led to the convening of the Constitutional Convention. Any major act of public disturbance or lawlessness could only be dealt with by the relevant state. If they chose not to do so, for whatever reason, then the federal government could do little or nothing. This is precisely what happened in the case of Shay's Rebellion. So it was felt necessary by Federalist delegates at Philadelphia to establish a central government with a police power capable of getting a grip on domestic disorder.
To the anti-Federalists, this once again smacked of tyranny. The United States had been founded on the basis of armed rebellion against a centralized government. The people sometimes needed to resort to direct action in order to stand up for their liberties as American citizens. The war may be over, they believed, but the principles on which it had been fought were still as relevant as ever, and must be protected. Whatever the value of such principles in theory, in practice it's difficult to see how an anti-Federalist system of government would have been able to stop the spread of serious public disorder from one state to another, thereby endangering the safety of the Republic as a whole.
In relation of foreign affairs, the anti-Federalists might have found it a little hard going without the existence of a strong central government. Under the Articles of Confederation, it was virtually impossible to formulate a distinct American foreign policy. The United States would need to gain the approval of all states before any treaty with a foreign state could be ratified, and this was a tall order indeed. The radically decentralized system of government favored by the anti-Federalists would've made it impossible for the United States to speak with one voice on the international stage.
The American economy would've been a whole lot different under the anti-Federalists too. They tended to represent agrarian interests, especially in the South. They deeply resented the close relationship that the Federalists had developed with commercial and banking elites, which were perceived by anti-Federalists as a threat to the economic interests of farmers and other landowners. As well as general concerns over too great a concentration of power at the federal level, this helps to explain why anti-Federalists were so implacably hostile to the establishment of a National Bank. Theirs was the Jeffersonian ideal of a republican democracy based upon a wise, disinterested class of landowners acting selflessly for the good of the country. Had their vision prevailed, the United States would not have developed along the lines of a modern, industrialized economy.