Politicians should let justice take its course over alleged crimes by British soldiers in Iraq.

Allegations that crimes were committed by British troops serving in Iraq from 2003 to 2009 need to be investigated, the prime minister said last month. But, Theresa May continued, ‘we do need to make sure… there isn’t an industry of vexatious allegations coming forward’.

The word ‘vexatious’ seemed to be aimed against Public Interest Lawyers, the firm run by Phil Shiner that ceased trading at the end of August. But, as the prime minister’s remarks suggested, there is a great deal of confusion here. How can we know whether allegations by Shiner’s clients are vexatious unless they are investigated?

Civil claims for compensation are a matter for the Ministry of Defence. But the job of sifting through alleged criminal offences was given to the Iraq Historic Allegations Team (IHAT). Based on the edge of Salisbury Plain at a military base built in 1912 for the Royal Flying Corps, IHAT is led by Mark Warwick, a former detective chief superintendent in the Metropolitan Police. He runs a force of nearly 130 investigators, some of them former police officers. Their work is regularly audited by Mr Justice Leggatt in the High Court.

By the end of August, IHAT had received allegations of potential criminal activity involving 3,368 potential victims. Initial complaints were quite detailed. But others received from 2014 onwards had little or no supporting evidence.

After sifting out duplicates, as well as incidents falling short of an offence and those where the difficulties of interviewing witnesses in Iraq would be disproportionate to the crime alleged, IHAT was left with 325 allegations of unlawful killing and 1,361 other alleged offences, ranging from serious sexual assault to common assault.

Last month, the senior Royal Navy Police officer at IHAT referred the cases of a serving major and two soldiers (one retired) to the Service Prosecuting Authority. They were suspected of involvement in the death of Saeed Shabram, 18, who drowned in the Shatt al-Arab waterway near Basra in May 2003. It is claimed the Iraqi civilian was forced into the water at gunpoint by peacekeepers who were trying to deter looters.

The major, who has not been named, has said that Shabram was pursued by an angry mob and troops had tried to rescue him. It is for the director of service prosecutions, Andrew Cayley QC, to decide whether any of the three should face manslaughter charges. The MoD has already paid Shabram’s family £100,000 in compensation.

The officer’s solicitor, Hilary Meredith, said her client had been investigated twice before – once as part of a wider official inquiry and once by the Royal Military Police. In 2006, the Army Prosecuting Authority decided not to bring the major and the two soldiers before a court martial because it believed there was no realistic prospect of a conviction.

IHAT told me: ‘We are of course aware that the incident was investigated previously, but were obliged to consider the case again because the courts ruled that such investigations were not sufficiently independent of the army chain of command. In our investigation into this case, we have uncovered new evidence which was not considered previously.’

When IHAT was set up in 2010, it was effectively run by the Royal Military Police and other members of the army’s provost branch. The Court of Appeal held in 2011 that this could give rise to a perception of bias, since military police might be investigating their own comrades. Ministers accepted this and replaced army investigators with members of the Royal Navy Police, supported by civilians.

Of almost 200 cases investigated by the end of August, 101 unlawful killing allegations and 91 ill-treatment allegations were found unsustainable. Two allegations – one of unlawful killing and of one ill-treatment – were sent to the Service Prosecuting Authority, which found insufficient evidence to proceed. One unlawful killing accusation was referred to the RAF Police for further investigation. One soldier was fined £3,000 by his commanding officer for ill-treatment.

IHAT told me that it could not say how many of the remaining allegations would need further examination. But it expected there would be around 250 cases under investigation in 2017 and 50 by 2018. Investigators expect to complete their work by the end of 2019.

Fatou Bensouda, prosecutor of the International Criminal Court, is keeping a close eye on IHAT. She announced in 2014 that she was reopening a preliminary examination into allegations that British officials were responsible for ‘war crimes involving systematic detainee abuse in Iraq from 2003 until 2008’.

But there will be no need for Bensouda to open a formal investigation – still less to bring charges – if IHAT can show it has examined these claims and taken action where necessary. That is why May was right not to scrap IHAT. And that’s why politicians should now let justice take its course.