As far as I can tell, the al-Malaki government and the Islamic Supreme Council of Iraq (the supreme council formerly known as SCIRI) which are the clients of both the USA and Iran are trying (again) to take control of the port from the anti Iranian British client Fadhila.
Michael Kamber and James Glanz write in The New York Times
In the weeks leading up to the operation, Iraqi officials indicated that part of the operation would be aimed at the Fadhila groups, who are widely believed to be in control of Basra’s lucrative port operations and other parts of the city. The ports have been plagued by corruption, draining revenue that could flow to the central and local governments. But the operation also threatens the Mahdi Army’s strongholds in Basra.
Fadilla is definitely outspokenly anti-Iranian. In The Guardian, Ghaith Abdul-Ahad wrote
"Fadhila, widely seen as backed by the British, split from the main Shia alliance in Baghdad after accusing it of having a sectarian agenda."
The risk is that the Irano-American assault on UK clients might trigger armed conflict between the Iran and US supported government and ISCI and the Iran liking US hating Mahdi army. So far, Moqtada al Sadr is calling for non violent resistance.
Hmm so is the US involved in a proxy war allied with Iran and against the UK ?
I'll ask Juan Cole.
He just links to Reuters where Aref Mohammed presents the fighting as mostly Iraqi government vs Mahdi army.
Update: McClatchy too describes the fighting as the philo-Iranian supported Iraqi central government against the Philo-Iranian Mahdi army. Leila Fadel and Ali al Basri at McClatchy have plenty of quotes from prominent figures on both sides.
They suggest that the battle is an electoral campaigning conducted by other means "The operation was portrayed by Sadrists and some analysts as an effort by Maliki to consolidate power in the south for the Islamic Supreme Council of Iraq before the election in October."