OLUSTEE, Fla. — There is an old saying in this state of seesawing sensibilities: The farther north you go, the farther south you get.

Florida’s northern counties have long seen the South as a kindred place — one that breaks the same biscuits, hunts the same deer and shares the same political bent. So around this tiny town 45 miles west of Jacksonville, on the edge of Florida’s largest and bloodiest Civil War battlefield, a Union incursion on sacred ground feels, to some, like reopening 150-year-old wounds.

“Old grudges die hard,” said John W. Adams, a former division commander in Florida for the Sons of Confederate Veterans. “And feelings run deep.”

Last year, nearly a century and a half after the Battle of Olustee, the Florida chapter of the Sons of Union Veterans of the Civil War made a request to the state parks department. It asked for permission to place an obelisk to honor Union soldiers (who lost the battle on Feb. 20, 1864) inside the three-acre Olustee Battlefield Historic State Park, the same patch of land that holds three monuments commemorating Confederate soldiers....