It's a worthy question, so let's examine it a little bit.

In my research I have been able to find plenty of discussion about the location of the Antonia Fortress (generally thought to be the northwest corner of the temple mount, but one scholar in particular thinks it's on the east), much debate about what it looked like (how large, what design), and some discussion about how many soldiers were stationed there (generally around 600, but as many as 10,000 could if the situation warranted it).

What I have been unable to find is historical examples of the soldiers of the Antonia intervening in disruptions. I was after what kinds of events prompted their show of force, how often such things happened, and how did they intervene. I have been unable to find anything. So if you have knowledge of or access to such information, I would be thrilled to see it and add it to my notes.

There is one such event in the Bible, but that's the only source I can find. In Acts 21.27, some Jews are making trouble for Paul. It is Pentecost, so Jerusalem is swollen with people. We can assume a large crowd at the Temple. This verse says "they stirred up the whole crowd." There is shouting and "the whole city was aroused" (v. 30). "The whole city was in an uproar" (v. 31), and the Roman commander acts (v. 32).

Even just a cursory reading of the cleaning of the temple shows a very different kind of scene: one man overturning an unknown number of tables, after which there is dialogue about what authority he claims to do such things.

If that's all we have to go by, it's very possible to see that the Romans may not have mobilized in that kind of situation.

What do you think?

I am talking about the cleaning of the temple. And I have a really hard time believing that the garrison of soldiers stationed there to quell any sort of disruption would simply discount some random Jew shouting and flipping over tables as street theatre.

“Stay thine gladius Marcus, me thinks that Jew may simply be engaging in a whimsical farce”

You'll have to be more specific. Jesus was not perceived by Rome as a seditionist. Even at His trial, Pilate found nothing wrong and no reason to crucify Him. Jesus interacted with a centurion at one time, and it seemed to be a friendly exchange.

Since you seem to be talking about the Temple Mount itself, it's true that a garrison of soldiers was stationed at the Antonia Fortress at the corner of the Temple Mount to be able to quickly quash any Temple Mount problems.

The only event that I might assume you're talking about is Jesus's cleansing of the Temple. None of the Gospel writers present it as a violent coup kind of event, but instead symbolic street theater as a prophetic message. It is often thought to have taken place under the covering of the Royal Stoa, at the completely opposite end of the Mount as the Antonia. It's hard to know how much of an actual disruption it was, in contrast to a seditionist act. In John's account, the Jews stick around to discuss his action with with him (Jn. 2.18-20).

I would like to discuss this further with you, but you'll need to clarify what you're talking about. Thanks.

We know that the temple would have had its own contingent of Roman guards on-hand specifically to ensure that nothing interrupted the commerce of the temple cult (since Rome had a direct financial relationship with the temple cult)

There was even a sign in the temple warning that anyone who interrupts the temple commerce would be arrested/killed (we still have a piece of it)

So why did the Romans who showed no hesitation to violently put down Jewish sedition in the past, not immediately murder/arrest Jesus?