According to President Barack Obama and the Pentagon, five countries — Bahrain, Jordan, Saudi Arabia, Qatar and the United Arab Emirates — joined the United States in this week's airstrikes in Syria. It's a staged family portrait: five Sunni governments united against extremism. Obama says the coalition shows that "people and governments in the Middle East are rejecting ISIL and standing up [for] peace and security."

Backstage, the reality is messier. The five official partners are mostly for show. In terms of firepower, they delivered less punch than the United States did. Qatar didn't participate directly, and it's already complaining that the bombardment is unfair and won't accomplish anything. Meanwhile, the United States is hiding or downplaying the involvement of other countries whose complicity, if acknowledged, might do more political harm than good.

Start with Syria. The United States can't admit to working with Syrian President Bashar Assad because he's the dictator our proxy ground forces, the Syrian rebels, are trying to overthrow. In reports, the Obama administration denied it had given Assad advance warning about the strikes. Yet we managed to hit at least 50 targets in three parts of the country — including the west, where Assad has substantial air defenses — without a flinch from the Syrian military. The regime didn't even deploy active radar against us. That's amazing. A little too amazing.

Unfortunately, Syria didn't play along with the no-warning story. The regime disclosed exactly what happened: U.S. Secretary of State John Kerry sent a letter to Syria's foreign minister via Iraq's foreign minister. The same message was delivered to Syria's U.N. ambassador. It specified where the strikes would occur.

Tuesday morning, the administration adjusted its story. "We did not coordinate with the Assad regime," said the Pentagon's spokesman. "While the United States did inform the Syrian regime through our U.N. ambassador of our intent to take action, there was no coordination and no military-to-military communication." The Pentagon defines coordination with the same precision Bill Clinton applied to the term sexual relations. In the case of the Islamic State, it depends on what the meaning of IS is.

The United States also denies coordinating with Assad's ally, Iran. An official U.S.-Iran partnership would alarm everybody: the Iranian right, the American right, the Gulf states, and all the Sunni tribes and fighters we're trying to cultivate in Iraq and Syria. Accordingly, Kerry excluded Iran from last week's conference of countries united against ISIS. But under the table, we're flexible. Iran's leader, Ayatollah Khamenei, says the U.S. undersecretary of state for political affairs requested Iran's help in a recent meeting with Iran's deputy foreign minister.

The Associated Press says the two countries have engaged in "back-room contacts about cooperation for weeks." Reuters adds that "in private, Iranian officials have voiced a willingness to work with Washington," at least in a nonmilitary capacity. "Washington's preferred dynamic," according to Reuters' sources, "is for Tehran to work separately toward the goal of defeating Islamic State while the two countries seek to 'deconflict' their activities." In Iraq, for instance, the United States has tacitly accepted Iran's arms shipments to Baghdad.

This week an Iranian official told Reuters that the United States discussed its Syrian military plans with Iran three times before going in, including a notification shortly before the attack. The official said the United States assured Iran we wouldn't hit Assad's forces. When the State Department was asked about the Iranian account, it fell back on Clintonian technicalities: "We communicated our intentions, but not specific timing or targets, to the Iranians. As we've said, we won't be coordinating military action with Iran."

The ally no one wants to acknowledge is Israel. That would play into ISIS propaganda, which frames Obama as the "mule of the Jews" and Saudi rulers as "guard dogs for the Jews." In the first Persian Gulf War, we used Israeli intelligence but didn't advertise it, lest we offend our Arab allies. Two weeks ago, Prime Minister Benjamin Netanyahu said of Israel's contributions to the anti-ISIS coalition, "Some of the things are known; some things are less known." An anonymous Western diplomat said the United States was using Israeli satellite images, "scrubbed" of their Israeli traces, to show its coalition partners damage from strikes against ISIS in Iraq.

No such role has been acknowledged yet in Syria. But the Obama administration began its surveillance flights over Syria only a month ago. In all likelihood, Israeli satellite coverage was even more thorough and useful in Syria than it was in Iraq. Last week, a senior Israeli military officer told reporters that Israel was passing intelligence about ISIS to the U.S. and that "if Israel has intelligence on ISIS targets in Syria — and are asked to share it with the international coalition — I believe that we would cooperate."

The biggest mystery player is Turkey. Officially, it has refused to join the military campaign. Cynics think this is part of a deal that freed Turkish hostages from ISIS custody a few days ago. But Turkey's policy already has a loophole: It forbids the United States from launching strikes from its territory against ISIS, except for unarmed drones. In an interview Monday, Turkey's president worried that bombing ISIS would leave the job "half done." Somebody has to deal with the social and political consequences, he argued. Maybe that's the role Turkey will adopt, at least officially.

Other countries, for various reasons, are limiting their participation or keeping it quiet. ISIS has beheaded one British hostage and has threatened to kill another. An ISIS affiliate in Algeria is threatening to do the same to a French hostage. It's not surprising that both countries, for now, are officially staying out of the Syrian half of the war. But it would be pretty shocking if Kerry and Britain's foreign secretary, in their meeting on the day of the airstrikes, didn't discuss what was about to go down.

When you see a list of the countries fighting ISIS, remember, it's not a coalition of the willing. It's a coalition of the partners our government is willing to divulge.

- By William Saletan