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In one case, nearly 200 illegal Malaysian workers were deported, fled the country or stopped at the border following a six-month investigation into Auckland's construction industry, the Weekend Herald revealed in February.
Investigators from Immigration New Zealand targeted 10 companies to disrupt the flow of illegal workers from Malaysia, who were responding to advertisements on social media.
Operation Spectrum raided construction sites around Auckland and detained and deported 54 people who were working unlawfully as plasterers, painters, tilers and carpenters.
But instead of being deported, illegal workers with construction skills might now be granted visas to work on the KiwiBuild project.
The idea has been raised with the Minister and discussed within Immigration New Zealand, according to several sources.
This was downplayed but not denied by a spokesman for Mr Lees-Galloway.
"We're not considering that right now. The Government will have some proposals for consultation coming out in the near future relating to construction industry and we'll look forward to hearing views from the public and business community," the Minister said in statement.
The proposals are unlikely to be made public until after the Labour-led government's first Budget is announced on May 17.
There were 747 people unlawfully in New Zealand deported in the 2016/17 financial year, as well as another 1437 who left "voluntarily".
But it is unknown how many had building or construction related skills.
As part of its election campaign, Labour - as well as New Zealand First and the Greens - promised to cut net migration.
A "KiwiBuild" visa was also discussed, where residential building companies could get a three-year visa for skilled migrants as long as they trained a local apprentice as well.
Michael Woodhouse, the National Party immigration spokesman, said Labour would be "hypocritical" if it allowed unlawful builders to stay in country after campaigning to cut immigration.
"There is no good reason to legitimise a person's immigration situation just because the construction industry is short of people," said Woodhouse.
"Amnesties don't work. That leads people to think if they if they stick around long enough, they'll eventually get a valid visa. That causes more unlawful migrants, not less."
Operation Landing
The Herald on Sunday has also obtained an MBIE briefing about a sting on residential building sites across Auckland.
Operation Landing, which involved multiple government agencies, targeted 152 new homes being built in the North Shore and south Auckland over a two-week period in 2016.
Among the key findings was limited supervision and oversight by qualified tradespeople, leading to poor building practices.
Four out of five building sites visited did not have a "licensed building practitioner" - who can carry out weathertightness work - present.
Investigators found problems with supervision of plumbing and gasfitting, including unlicensed work, on nearly 30 per cent of sites.
Consented building plans were missing from up to 80 per cent of sites, while only 42 per cent of sites were meeting their employment obligations.
The workforce on each of the 152 building sites were broken down into "major ethnicities" with nearly 65 per list listed as "Asian/Chinese".
It was "very likely" a number of building sites were involved in the cash economy and not paying tax to Inland Revenue.
The conclusion of Operation Landing was "that the Auckland new residential build sector has a number of regulatory issues and challenges".
"Notwithstanding the credible job undertaken by Council building inspectors to ensure the new build residential construction meets the standards set in the Building Code, a number of significant issues exist ... which potentially undermines the quality of the build," the MBIE report author Mike Hill wrote.
"There is clear evidence that the consents for new dwelling builds are rising each year, which requires more workers.
"The source, capability, qualifications, supervision and management of existing workers currently present problems. This will increasingly be more challenging in the years ahead."
