The next stage in a battle over furloughs between Gov. Jerry Brown and California’s state engineers’ union is set for next month when the two sides will argue the case in a San Francisco appellate court.

At stake is an estimated $12 million in back pay that a trial court said the state owes to roughly 13,000 members of Professional Engineers in California Government. The union successfully argued that the governor in 2011 furloughed those employees two days longer than the Legislature had authorized.

The union also persuaded Superior Court Judge Steven A. Brick that about 250 of its members working on toxic-waste remediation at military bases should not have been furloughed at all. Unless the lower court’s decision is overturned, those workers would receive back pay for up to 70 days of unpaid time off mandated by the state from 2009 through 2011.

The 1st District Court of Appeal has set July 9 at 9 a.m. to hear arguments for and against the lower court’s ruling. The appellate court justices have 90 days after hearing oral arguments to issue an opinion.