Boarding dilemma, "Obey instructions".....
 

Round trip flight on SW but each leg on a separate itinerary. Two passengers on each itinerary (couples).
On the way back checked in using the smart phone app - it gave the boarding pass . (Did the same thing on the forward leg too).

At the airport checked in a bag and therefore printed boarding passes for the itinerary - it gave A4_ and B5_ boarding position.
So stood on A4_ and scanned the boarding passes. The gate agent noticed that the 2nd is B..... and asked the female to not to board. (Agent already removed portion of the boarding passes as if boarding) The male passenger told the agent that it is the same itinerary. Agent said go to customer service - when asked for agent's name and pulled the phone out, the agent closed the terminal Gate and ran away.

Agent came back with another person and introduced him as the supervisor. This guy took the passengers away claiming "refused to obey instructions/orders" .... said you can not go in that flight. Later, he took the passengers to another terminal Gate where another gate agent present. Repeated the same thing. Then taken as the last set of passengers on the original flight - a different agent wanted to scan the boarding passes - agent was told it was scanned already. The supervisor said that she (previous agent) removed you from the flight.

Passengers were made embarrassed (in front of the everyone) - essentially told that you are in shackled and gagged when you hold SW boarding pass. Is this how it is? What options are there in pursuing the grievance?

Update:
Not Terminal but Gate

(NOTE - I am assuming when you said you went to another "TERMINAL", you mean a computer terminal and not another building)

They are two different boarding positions, clearly. A4 and B5.

Pulling out a phone and CONFRONTING just about any person is looking for trouble.

Based on even one side of the story, when they told you that B5 could not board yet, you should have just gone with it and not started this whole thing in motion.

I'm not sure I completely understand, but it sure sounds like someone with a B boarding pass tried to board with her mate, who was assigned to the A boarding section. She was asked to wait until her turn came around, but they persisted and complained? And somehow their ensuing hassles and embarrassment were Southwest's fault?

I'm confused too. Passengers appear to have broken fares (segments on different tickets) but both passengers are on the same reservation (same itinerary?). It would help to know whether the male has elite status or purchased early boarding or something to explain why two traveling together have such different numbers.

However, couldn't he just board in his assigned spot and save a seat for her?

From what I can surmise, pax tried boarding out of order, was caught, and made a scene that disrupted boarding. Lucky they got to fly at all. Lesson: don’t do this again. Oh...and really...”shackled and gagged”?

You were witness to this felony? Third-person narrative (e.g, "the male passenger") is unconnected scuttlebutt.
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If the agent "closed the terminal and ran away" boarding was suspended. No minor thing. But the airline should turn away anyone going to Defcon-1 over a long-recognized rule. A calm(er) recitation of events supported by witnesses is your lone recourse.

By your own admission, a passenger tried to board out of order. The passenger was caught, told not to board, and made some kind of a fuss.

Whatever the fuss, it resulted in the passenger being offloaded and eventually permitted to board.

What is missing from this narrative is exactly what the passenger said and did when caught. Although you have not said so, it sounds as though the "passenger" was either you or your companion or you somehow witnesses all of this.

Whether offloading the passenger was an overreaction is entirely dependent on exactly who said what and when and you have not supplied that detail. WN does not offload passengers lightly, so without detail, one has to presume that the fuss was sufficient.

Not sure how we get to "shackled and gagged". When people use hyperbole such as that without supporting facts, it tends to cast a lot of doubt on their story.

when you pull the phone out (presumably to film the agent?), I can assure you that bad things are coming your way.

Follow the rules. If the passengers can't deal with boarding separately, then they should board at the B position together. End of story.

Extremely confusing narrative, which is likely incorrect/incomplete anyway. I’d be curious to know how one person got A4 exactly.

Do what you are told. Problem solved. Airline personnel trying to get flights out do not have time to deal with your silly nonsense. One gets on at A4, the other gets on later, what’s the problem?

I am happy to get B5.

This is a very confusing post but as best I can tell, two people were on the same itinerary and thus you'd think they'd receive consecutive boarding positions when they check in. For whatever reason, one person received A4 and one person B5. B5 tried to board with A4. Either these people thought it was ok for two people on the same itinerary to board with the better boarding position (which is simply not the case), or they willingly tried to cheat the system. Either way, they were caught and one passenger was told to board with B5. This should have been the end of the story. Instead, they apparently made a stink (including asking for the gate agent's name, and beginning to film the encounter via cell phone) and the result is this post.

If what I've typed above is correct, then I 100% agree that the belligerent passenger(s) should have not been permitted to board. Depending on the severity of the outburst, the person(s) possibly should have been preventing from flying. Without having seen the outburst or the gate agent's reaction to the whole situation, it is hard to judge the severity of the situation.

could be one was BS and the other companion pass, that would account for the different boarding positions.

Your village called. They said could you return home immediately as they are missing you.

The OP had an underscore after the numbers, so I think they were A40-something and B50-something, rather than one being in BS at A4.

Totally agree.

I wonder why someone would even do that - over a "boarding order issue".

There must be more to this story.