The PSA later confirmed the District Courts in Whangārei, Auckland, Manukau, Hamilton, Rotorua, Palmerston North, Dunedin and Nelson will see workers walk out.
The High Court at Auckland and Christchurch's Justice and Emergency Services Precinct will also be affected by striking staff.
Meanwhile, a PSA rally at Parliament this afternoon coincides with New Zealand's first Public Service Day.
The ministry's chief operating officer, Carl Crafar, said the "situation remains fluid".
"We are doing our best to manage the impact on our customers and our people," he said in a statement.
"The ministry remains open and committed to reaching a negotiated settlement and we're ready to meet with the PSA at any time."
On Monday, the Employment Court at Auckland ruled against the ministry as it sought to at least temporarily end the short-notice strike action.
Crafar earlier told the Herald the ministry had "serious concerns" about the health and safety risks of the lightning strikes.
"We have taken this action as we feel we have been left with no other choice to ensure the safety of our staff, the public and the judiciary," he said.
"We consider the PSA's decision to strike with only 30 minutes' notice at crowded and busy courts to be unlawful, unsafe and irresponsible. We will do everything we can to protect everyone working in or visiting our courthouses."
But the Employment Court's decision sees no barrier to further strikes with a 30-minute notice period.
Bargaining on a new contract with the PSA has been continuing for several months.
The ministry has offered a 5 per cent increase over two years, in line with other agreements in the public sector.
However, the PSA initially sought an increase of more than 13 per cent and also wants to close the gender pay gap.
Crafar said the ministry has a budget of 3 per cent of its annual salary bill for pay increases this year and 3 per cent for next year.
"The ministry has also offered an additional $750 one-off payment to PSA members for 2018," he said.
"The PSA initially presented a pay claim which added up to more than 13 per cent during this period, more than double the ministry's budget. They have subsequently reduced their claim to 11 per cent."