While foreign leaders and publics appear to have taken a largely relaxed view to President Barack Obama’s re-election—likely foreseeing, and being prepared to accept, considerable continuity—the future of US foreign policy over the next four years in fact seems quite uncertain.

America will face critical choices and unexpected challenges in an environment of domestic economic anxiety and political tension. Succeeding will require skills that the president has not yet convincingly demonstrated.

A Divided America

The essential departure point in looking ahead is recognizing that despite Mr. Obama’s solid victory in the Electoral College, the popular vote was relatively close—and America remains deeply divided. Moreover, notwithstanding extensive commentary suggesting that the outcome was “inevitable,” the election was winnable for Republicans in general and Governor Mitt Romney in particular had Republican Party and Romney campaign leaders made different strategic and tactical decisions.

For example, mistakenly attempting to replicate the Democratic Party’s long and bloody 2008 nomination fight in the belief that it would “strengthen” the Republican nominee was clearly damaging to Governor Romney by forcing him to run as a “severe conservative” for much longer than might otherwise have been necessary.

This cemented many voters’ views of his positions on immigration and divisive social issues—perceptions that otherwise may have been more easily erased when the “Etch-a-Sketch” moment arrived after the Republican National Convention—and undermined his wider appeal in the general election.

Republican Party leaders have exacerbated this problem over many election cycles by focusing campaign efforts on accommodating and even exciting extremist activists rather than educating and managing them. From this perspective, Republican losses in the 2012 Senate races may well be more damning than Governor Romney’s defeat—especially in contrast with what appeared possible just one year ago.

Can the Fiscal Cliff Be Avoided?

Still, the fact that Republican Party candidates and leaders failed to seize important opportunities does not mean that Americans have rallied behind President Obama—far from it. And notwithstanding conciliatory statements from all sides as the country approaches the so-called “fiscal cliff,” a bitter partisan battle over taxes and spending may not be far off.

President Obama’s success in working with Republicans and Democrats in Congress to make a workable deal could well define the tone for his second term. Unfortunately, the president’s track record in handling his original stimulus bill and health care legislation do not inspire optimism.

Looking a little further ahead, whether and how the United States avoids going over the fiscal cliff will have significant implications for US foreign and security policy, starting but not ending with the defense budget. President Obama’s success in addressing the federal budget—and generating significant economic growth—will also powerfully affect his effectiveness at home and therefore abroad, where world leaders will gauge his capabilities based in part on his domestic performance.

As he deals with these enormous and complex challenges, President Obama will also have to restructure most of his foreign policy and national security team. According to press reports, Secretary of State Hillary Clinton and Secretary of Defense Leon Panetta each plan to step down relatively soon, though both appear prepared to remain in office until a successor can be confirmed by the Senate. The unexpected resignation of CIA Director David Petraeus only adds to the task.

New Cabinet

A major personnel transition of this nature raises two categories of important questions. First, will Mr. Obama’s new team be effective? Secretary Clinton has often had greater credibility among Republicans than President Obama—will her successor? Similarly, Secretary Panetta has extensive experience serving in and working with Congress; will the next Secretary of Defense have similar capabilities? How will Mr. Obama’s low-profile National Security Advisor, Thomas Donilon, work with his new colleagues?

And second, how will the administration’s new personalities shape the president’s priorities? While Mr. Obama’s broad objectives appear unlikely to change, new people can have a major impact on which goals receive priority treatment and how they are implemented.

Here, even lower-level personnel changes can have significant consequences—and when a secretary of state or secretary of defense leaves office, under secretaries and assistant secretaries who have been close to them, and therefore uniquely effective, often go too. As a result some initiatives may lose their day-to-day bureaucratic interpreters and champions.

The final consideration in thinking about the next four years in US foreign policy is what others will do. This has three components: what role Republicans will seek in shaping policy, how foreign governments will react to developments in the United States, and what the next unexpected crisis will be.

With a majority in the House of Representatives, Republicans are certainly in a position to affect important decisions through the power of the purse and also have a visible platform from which to present an alternative perspective—if they can succeed in defining one more appealing than the Bush-era platitudes that did little to help Mr. Romney. Likewise, the administration’s policy will not be divorced from the actions of key global players, whether friends like Germany and Japan or rivals like China and Russia.

Unfortunately, the administration does not have an impressive record in dealing with surprises—something most clear in its responses to the turmoil in the Middle East, whether in Egypt, Libya or Syria, and probably due in part to the administration’s lack of a clear overall strategic framework for US foreign policy. With this in mind, President Obama and his team would do well to expect the unexpected and to think about potential problems and options in advance. Those outside the United States seeking to predict American policy in Mr. Obama’s second term should be prepared for possible surprises too.