TMZ Sports has obtained new footage of the Rex & Rob Ryan bar fight in Nashville -- which shows Rex cussing out a bar patron and then spilling a margarita on the guy.
It all went down Sunday at Margaritaville -- you can see Matt Havel chatting up the ex-NFL coaches and another man at the Ryans' table ... until Havel said something that set Rex off.
Unclear what Matt said ... but Rex blows up at the guy. Rob shoves him. And while Rex grabs the guy's drink to move it to another table, someone bumps his elbow ... and Matt gets splashed with booze.
The fight quickly escalated ... with Rob eventually grabbing Matt by the neck.
Havel later filed a police report accusing both brothers of simple assault.
That said ... props to the singer who continued to play a pretty solid rendition of the Kenny Rogers classic, "The Gambler" while the scrap broke out.
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