Mr McKenzie's review found that Overseas Investment Office (OIO) investigators had discovered one "relevant matter" during internet searches of the company's background. However, the discovery was not included in a final report to ministers.
Mr McKenzie's review did not give any details about the "relevant matter".
He recommended that all relevant findings about prospective investors were passed on to the Government -- a measure which the OIO said it had already implemented.
Mr Cunliffe said the review appeared to be a "watered down" investigation, taking just a month and looking at only a handful of cases.
"The biggest recommendation is to provide Google searches to the Minister to help assess cases," he said.
"That is basic stuff and doesn't shore up the good character test. The changes must be much more comprehensive than that."
Labour wants the OIO to be better funded, saying that it is effectively rubber-stamping applications because it is under-resourced.
During his review, Mr McKenzie also discovered that the company's directors, Argentinian brothers Rafael and Federico Grozovsy, used "bearer" shares -- a form of shareholding which made it difficult to trace the owners and could easily be transferred to another party.
Mr McKenzie recommended that all foreign investors should have to convert bearer shares into registered shares or move them into "reputable" banks before they were given OIO approval.
It was not known how many successful OIO applications involved investors with bearer shares.
Ceol & Muir's OIO application was originally reviewed in April because of its links to Mossack Fonseca, the law firm at the centre of the Panama Papers. A day after it was cleared of any wrong-doing, Mr Cunliffe revealed its past prosecution for toxic pollution, prompting the review to be re-opened.
NZH