The confrontation occurred about 2 p.m. Sunday outside the Smart & Final in the 5200 block of East Ramon Road, just east of Highway 111, according to the Palm Springs Police Department.
Police said the two men were involved in an unknown dispute that escalated into a gunfight, culminating in both pulling handguns and “shooting at each other.”
Witnesses called 911 as shots rang out, and patrol officers converged on the location minutes later, but the two suspects had already fled, according to police.
No bystanders were hurt, and there was no evidence at the scene that either man had been shot, investigators said.
One suspect was described as heavy set, with a beard, wearing light-colored pants, a dark shirt and a gold chain. The other man was slender, bearded, with long hair wrapped in a bun, wearing a multi-colored shirt and shorts.
Anyone with information was asked to contact the police department at 760-323-8129.