Monday the Iraqi military launched its largest operation to date against the self-declared Islamic State (IS), also called ISIS, to retake control of the city of Tikrit. Alongside the Iraqi military the coalition fighting IS in Tikrit includes Kurdish and Sunni tribal forces, but it leans heavily on Iranian-backed Shia militias and reportedly includes a contingent from Iran’s Revolutionary Guard. The urgent question now as the battle against IS intensifies is whether any U.S. policy to defeat IS in Iraq can achieve its aim without ceding the country as a base for Iranian expansionism.

Critics who regard President Obama’s regional policy as aiming for a grand détente with Iran have frequently argued that the current approach undermines attempts to counter IS. The bargain for making a deal with Iran, these critics say, has allowed Iran a free hand to assert dominance in Iraq, Syria, Lebanon, and Yemen.

Exhibit A for this line of thinking is Iran’s cultivation of proxy militias in Iraq, principally in the Badr Organization, Kata’ib Hezbollah, and Asa’ib Ahl al-Haq, These militias have not just been active on the frontlines in Iraq but also have arguably played the leading role in all major offensives to retake territory from IS, with Iranian Revolutionary Guards Corp figure Qassem Suleimani helping to direct operations.

The Badr Organization in particular, with its control of the Interior Ministry, seems poised to become Iraq’s version of Hezbollah. Last fall, the group launched “Operation Ashura” to clear out Jurf al-Sakhr to the south of Baghdad. Under Suleimani’s guidance, the operation employed a successful strategy of amassing vast militia manpower operating under cover of U.S. airpower.

A virtually identical tactic is now being implemented in the offensive to capture Tikrit except this time with Iraqi fighter jets taking the place of U.S. airpower.

Since these proxy militias frequently engage in ethnic cleansing against Sunnis and answer directly to Iran, they bolster IS’s narrative that it is defending Sunnis against a sectarian government, arguably undermining any attempt to roll back IS. Besides, reflecting Iran’s own anti-American ideology, they also promote a narrative that the U.S. is behind the IS phenomenon, further undermining U.S. influence in Iraq to the benefit of Iranian expansionism.

There is much to be said in favour of these arguments. Since the fall of Mosul to IS in June 2014 and the call to arms issued by Iraq’s most senior Shi’a cleric, Ayatollah Sistani, militias that are ideologically aligned with Iran (“Khomeinist”) if not actual proxies have proliferated most, with many new brands emerging beyond the three mentioned above. A considerable degree of overlap exists between these new groups. For instance, one commander I interviewed is simultaneously involved with two recognizably Khomeinist militias: Kata’ib al-Imam al-Gha’ib (a “Hezbollah” brand) and the Mujahideen of Iraq Brigade, the “military wing” of the Nasrallah Islamic Movement (named after Hezbollah leader Hassan Nasrallah).

Beyond the question of Iranian influence, those who defend the “Popular Mobilization” trend as a military necessity tend to downplay the more general negative consequences of militiafication. There is little reason to accept former U.S. defence official Douglas Ollivant’s contention that the militias “will either return home or be regularized by the central government in some way” once the IS threat is dealt with. Militias also create an atmosphere of lawlessness and criminality regardless of the sectarian issue. Indeed, some of the militias themselves have acknowledged the problem of kidnappings and stealing in their name, including a Khomeinist militia known as Kata’ib Ruh Allah.

The complaints about Iran’s expanding influence in Iraq are valid but they raise an important question that has yet to be answered: How exactly do you curb Iranian influence at this stage when its forces dominate in Iraq? The usual line here is to say that the U.S. needs to stop abetting the Iranian proxies through airstrikes and arms provisions to the Iraqi government. But going back even conditionally on these measures simply creates a bigger military vacuum for Iran to fill. At the same time, Iranian proxies are undoubtedly spearheading most new offensives by government forces against IS and at least some of the new weapons shipments intended for Iraqi security forces are likely to end up in the hands of Iranian proxies.

One also hears calls for new U.S. engagement in Iraq, but there is no honesty about the scale of commitment that would be required. If the goal is to rebuild Iraq’s conventional security forces as an alternative to the militias, then the reality is that there will have to be tens of thousands of ground troops, deployed for a number of years and not only willing to train these new forces but coordinating with them in combat missions. Yet even such a massive commitment—tried once before in recent memory—has no guarantee of success. Such an approach is also politically unfeasible due to American war weariness and scepticism of any mission with shifting goalposts. Further, a large-scale American ground presence risks fueling further support for IS, the possibility of infiltration of rebuilt army brigades by the Iranian proxies, and open warfare between the proxies and U.S. troops.

Notions that the U.S. should focus only on cultivating Kurdish and Sunni allies are also unrealistic. Kurdish forces alone are unable to dislodge IS from its main strongholds, and Sunni locals have good reason for concerns about treatment at the hands of Kurdish forces.

The strategy employed during the last U.S. war in Iraq, employing Sunni tribal groups to lead the fight against IS, has its own problems.

With supposed Sunni allies, the biggest question remains of who is out there for the U.S. to approach.

Sunni insurgent actors like the Ba’athist Naqshbandi Army find themselves severely weakened, having lost out to IS in all major towns and cities outside of government control. Local Sunni forces that are actively pushing back against IS in Iraq’s Anbar province are in fact already working with the Iraqi government and the militias but have been unable to dislodge the group.

On the political axis, Sunni politicians are more lacking in credibility among their constituents than ever.

Simply put, there are no viable “third-way” Sunni actors who reject both the government and IS.

Concern has been expressed that the U.S. “risks” losing Iraq to Iran in the fight against IS, but it is probably more accurate to say the U.S. has already lost Iraq to Iran. No good options seem to exist, and the expansion of Iran’s sphere of influence may well have to be accepted as an inevitable consequence of the original decision to invade Iraq and remove Saddam’s regime from power.

Update: This story originally stated that U.S. airpower was supporting the Iraqi offensive against the Islamic State in Tikrit as it had done in earlier operations in Iraq. After the story was published the Pentagon announced that it was not providing airpower in support of the operation. The article has been updated to reflect that change.