Suppose Sen. Barack Obama (Ill.) wins the Democratic nomination and picks Republican Sen. Chuck Hagel (Neb.) or Independent New York Mayor Michael Bloomberg as his running mate. Or, suppose Sen. John McCain (Ariz.) wins the GOP nomination and picks Independent Democratic Sen. Joe Lieberman (Conn.) as veep.

[IMGCAP(1)]Suppose even further that, over this year’s holidays, Speaker Nancy Pelosi (D-Calif.), Senate Majority Leader Harry Reid (D-Nev.) and President Bush all resolve that next year they’ll really try to live up to the pledges they all made in early 2007 to work across party lines to — as they all said — do the problem-solving work voters elected them for.

Is it all fantasy? Perhaps it is, given the hyperpartisanship of contemporary politics. Yet, every poll on the subject indicates that Americans are fed up with their politicians’ incessant tribal warfare and inability to address problems everyone agrees are becoming more serious from inattention.

If the two parties’ presidential nominees reached out across party lines to pick their running mates — Obama and McCain seem the likeliest to do so — it would serve as dazzling notice that times were changing.

It would be even more astounding if Congressional leaders and Bush could decide that, instead of repeating the dismal, few-achievements record of 2007, they’d resolve to solve at least one major problem in 2008 — say, pass tough but compassionate comprehensive immigration reform.

Over the holidays, America’s political actors — and observers — would do themselves and the country a favor by reading Ron Brownstein’s new book, “The Second Civil War,” whose subtitle begins to tell it all: “How Extreme Partisanship Has Paralyzed Washington and Polarized America.”

Brownstein, formerly with the Los Angeles Times and now political director of Atlantic Media Co. publications, vividly describes the historical origins of “hyperpartisanship,” a term he borrows from a sometime practitioner of it, former Republican National Chairman Ken Mehlman.

More importantly — Brownstein eloquently laments the consequences of the disease and offers some fascinating remedies, some derived from former President Bill Clinton, whom he interviewed at length.

Brownstein doesn’t suggest picking vice presidents across party lines. Those are my radical imaginings — though they are derived from conversations with participants in presidential campaigns.

Brownstein has this right: America is the richest, most powerful nation on Earth, but its leaders can’t agree on a plan to reduce dependence on foreign oil, can’t balance the budget, can’t provide health insurance to a sixth of its population, can’t align its promises to retirees with its ability to pay the cost and can’t agree on strategies to combat Islamic terrorism.

Why not? Because solutions to these problems require bipartisan “grand bargains” that polarized politicians are unwilling to make.

“Our politics today encourages confrontation over compromise,” Brownstein writes. “The political system now rewards ideology over pragmatism. It is designed to sharpen disagreements rather than construct consensus. It is built on exposing and inflaming the differences that separate Americans rather than the shared priorities and values that unite them.”

Brownstein puts primary blame on conservative Republicans for the rise of “warrior” politics, especially former Speaker Newt Gingrich (Ga.) and House Majority Leader Tom DeLay (Texas), Bush and his former guru, Karl Rove, and their allies on talk radio.

But he observes that Democrats are catching up in hyperpartisanship, flogged on by MoveOn.org and leftist bloggers. Mainstream media, too, encourage conflict over consensus. And the public has become ideologically “sorted,” as well, making the GOP more conservative, Democrats more liberal and moderates torn.

Brownstein gives rather more credit to Clinton than I would as a model centrist. He was that on policy — the “Great Triangulator” —but his personal misdeeds, slipperiness and tendency to respond savagely to threats made him as divisive as Bush, the “Great Polarizer.”

But how can we end the war and engender vigorous, substantive debate that leads to consensus? Brownstein recommends that states banish closed primaries and allow registered independents to participate in picking candidates.

He also advises that political leaders look to a growing corps of cross-interest coalitions — such as the Business Roundtable, Service Employees International Union, AARP and National Federation of Independent Business — working to develop consensus solutions to problems such as health care and entitlement reform.

But the prime requirement is presidential leadership — a willingness to spend time with leaders of the opposition party, include them in policy deliberations, really heed their concerns and try to build electoral coalitions and Congressional support of 55 or 60 percent, not Bush’s 50-plus-one.

“Imagine … that such a president told the country that he would accept some ideas counter to his own preferences to encourage others to do the same. Surely such a president would face howls of complaint about ideological betrayal from the most ardent voices of his own coalition.

“But that president also might touch a deep chord with voters. … It has always been true that a president can score points by shaking a fist at his enemies. But a president who extends a hand to his enemies could transform American politics.” Amen. Think about it over Christmas.