Khloe Kardashian & James Harden -- Set Off HUGE Birthday Fireworks ... City Officials PISSED! (VIDEO)

1:00 PM PT -- We've now learned Khloe and James did NOT need a permit to set off the fireworks. They did need the Coast Guard to sign off on the safety of the display, and they got that.

Khloe Kardashian and James Harden threw their own personal 4th of July with a GIANT fireworks display that woke up thousands of residents -- and city officials are scrambling to find out HOW it got approved.

The thunderous fireworks show -- visible from multiple towns along the oceanfront -- went on for 15 minutes at the stroke of midnight. TMZ has learned the whole thing was a yacht party for Harden's 26th birthday, and it was all on the up and up.

Law enforcement sources tell us a permit was issued to James and someone in the Kardashian family. Khloe had bragged about throwing the party for her BF, but nobody expected this kind of over-the-top display.

In fact, there's a mini-conspiracy theory brewing. We've spoken to the L.A. department that normally issues permits for fireworks in the marina ... and they did not sign off, and never would for such a display on a Tuesday night. We're told the display was far enough off shore to be outside of city jurisdiction.

We also spoke to an LA. City Councilman who is angry and trying to figure out who exactly OK'd the fireworks. One law enforcement official told us ... in 5 years he's never heard of a permit being issued for fireworks like this on a non-holiday.

It really is their world, the rest of us are just living in it.