Gulf Arab investment in Iraq has swelled in recent years, signaling a new and widening rapprochement with Baghdad. Yet, while Iran and Saudi Arabia last year agreed to normalize relations, it is unclear whether Arab investment in Iraq—without stoking Iranian ire—is tenable. Riyadh and its Arab neighbors recognize that any move against their interests by Iran-backed Iraqi armed groups would undermine the investments that their broader strategy depends on.

In other words, the success of Saudi economic and political initiatives depend on not being seen by Iran and its allies as excessively interfering in Iraqi affairs. Riyadh’s growing economic stake in Iraq suggests that...