You can't have them, said general manager Graeme Wright. He was struggling to meet demand from his regular distributors.
What if I fly down and collect them, the persistent caller asked.
Sure, you can fly down if you want but you still can't have them, Wright responded.
Such is the demand for the delicacy dredged from Foveaux Strait, the world's largest wild oyster fishery.
Customers were queued outside the Barnes factory on a bleak stretch of Invercargill's wide Spey Street last week when the season opened.
Sales were frantic.
A total of 4500 dozen were sold in one day, going straight from the openers to the shop.
The price is $25 a dozen at source and as much as $65 at upmarket Waiheke Island in the north.
Culpeper's in Auckland managed to secure 500 dozen and ran an all-you-can-eat promotion at $175 a head. One diner scoffed seven dozen.
The parasite bonamia ostreae has destroyed the farmed oyster industry on Stewart Island and in the Marlborough Sounds but there is no evidence of it infecting the wild fishery, as was feared.
Wright, in his role for 19 years, has seen wildly varying seasonal conditions and complete closures in several years but is cautiously optimistic about the way this season is shaping, which opened on March 1.
An encouraging sign is strong oyster recruitment, the best since 2008, and low mortalities in survey tows conducted over the summer.
The oysters are infected with a less virulent strain of bonamia - exitiosus - which is not harmful to humans, and is at low levels this season.
Barnes Oysters, a co-operative of eight seafood companies, employs 18 openers, a loyal staff that return year after year.
They include 80-year-old Errol Ryan, still going strong.
A fast opener earns about $50 an hour.
Some are tradies taking a break from their regular work, others are seasonal workers and some like Errol Ryan cannot stay away.
Those long-term loyalties extend to distributors who may have been dealing with the company for decades.
The most celebrated was Frank Whiting in Auckland who died last year aged 101.
He still drove when he was 99 and was selling oysters at 100.
Despite the demand, the quota holders are determined not to over-exploit a precious resource and fish conservatively.
The Total Allowable Commercial Catch is again 15 million but last year only about 10 million were landed.
The season ends on Aug 31 but most of the 11 boats in the fishery have tied up by then.
A highlight of the season is the Bluff oyster festival on May 25 which has again sold out.