Play video content EXCLUSIVE TMZ.com

Things got downright HOSTILE on the Jersey Shore last night -- with NJ Governor Chris Christie getting into a heated verbal altercation right on the boardwalk ... and it was all caught on tape.

Gov. Christie (R-NJ) was hangin' with his family in Seaside Heights ... and had just ordered an ice cream cone when a passerby fired off some snide comments about Christie's policy on education.

Christie got PISSED -- and while clutching his cone, shouted back at the guy, "You're a real big shot ... you're a real big shot shootin' your mouth off."

The man shouted back, "Nah, just take care of the teachers!"

The comment only inflamed Christie ... who aggressively marched towards the guy and warned, "Keep walkin' away ... really good ... keep walkin'."

Soon after, someone in Christie's party put his arm around the 49-year-old and escorted him in the other direction.

FYI -- Christie, one of the prime contenders for the Republican Vice Presidential nomination, has recently signed legislation expanding public funding for charter schools ... a decision that has infuriated some who would prefer the money go towards struggling public schools.

Christie's temper has been making headlines after he called a reporter "stupid" and an "idiot" on Saturday for asking off-topic questions at a news conference.