The home franchise's technical adviser, Mark Foster, was banished to the stands for dissent, according to the website report.
The straight red card to Mulligan changed the complexion of the game but the visitors, Greatholder argued, were the better team before that.
"We dominated in every aspect of the game but the referee [Mirko Berishke] changed the course of the game."
Greatholder was adamant the former All Whites midfielder's tackle wasn't dangerous, something his crop of "honest boys" had verified at halftime and "everyone [Bay players] on the field thought that except the referee".
"From there on we clearly had no ball and we didn't react too well. It's tough."
Some whistle blowers, he felt, would have issued a yellow card "but that's what goes around nowadays when things are open to interpretation".
Greatholder said Corey Chettleberg's tackle in the 12th minute of the game "on top of the ball" was more incriminating than that of Mulligan.
"But that's football and they returned [after halftime]. They could smell blood and started piling forward and goals kept coming."
Charlie Henry opened the floodgates in the 47th minute in what Greatholder described as, "we gave them a horrible, horrible goal".
Bay goalkeeper Richard Gillespie threw a ball out to the field but it caught the back of Henry, who pounced on it, went around the gloveman and drilled it into the net to make it 1-1.
Henry Fa'arodo made it 2-1 in the 68th minute, Joel Stevens (78th) and Henry (83rd) followed before ex-Bay United striker Hamish Watson got into the act from the bench with goals in 86th and 90th minutes.
He didn't want to take anything away from Wellington because "they didn't make the ref's decision".
Having changed his formation at halftime, Greatholder said his men felt they could still salvage a draw, if not win. "We were angry because we were shafted, as it were, but there's no excuses ..."
Wellington coach Matt Calcott said the Mulligan ruling "was definitely a red card".
"If you jump in like that then you deserve to be sent off from the field," Calcott said, adding his team kept their discipline.
Preferring not to become embroiled in a "war of words in the media", he said Wellington played "great football".
"In the second half we were excellent but in the first half it was a physical battle and that's not our kind of game," Calcott said, believing a lot was riding on the game so it was "emotive".
He instructed his troops not to get involved in the second spell after numerous challenges and they had responded maturely. "We are playing from week to week for the next four weeks and we don't have any playoff or O-League goals."
The Bay host Wanderers Soccer Club in a early-round deferred match on Thursday before playing Waitekere United on Sunday at Park Island again.
With the Dragons winning 2-0 in the Southern derby, they have secured second spot on the ladder on equal points behind Auckland City, who Waitakere United pipped 1-0 yesterday.
The Bay slip to fifth place, two points behind Wellington on 19, but the former have played one fewer game and should go above them assuming they win on Waitangi Day.
