Tiger Woods was still so out of it 2 and a half hours after cops pulled him over for DUI ... he stumbled and mumbled his way through breathalyzer testing.

Police released video of Woods back at the station at 4:30 AM Monday. The woman tasked with testing him had to whistle and wave in Woods' face just to get him to open his eyes. Explaining the steps to take the test wasn't easy either.

Multiple times she has to tell him not to suck, but blow into the tube.

As we've reported ... Tiger ultimately blew .000 -- supporting his statement he was not drinking. However, this video -- just like the images of him attempting field sobriety testing -- makes it clear he had no business being behind the wheel. He says he had a bad reaction to combining prescription meds.

The sole light-hearted moment -- Tiger describing his hair color as "mostly brown and fading and ... brown." Even the cop had to laugh.