I’ve read (I think here) that “and that’s an order” is pretty much a TV/movie trope. What about asking for permission to speak freely, is that a trope too? If not, when would it actually be said?
I don’t recall ever doing or seeing this in my five years in the Navy.
I doubt there is a “rule” about this.
You are just asking the other person to not punish you for what you say next for being insubordinate.
Hopefully they will honor that casual agreement. I suspect most people would if they give permission to “speak freely.”
Kinda like telling a reporter something you say is “off the record.” There is no law or rule about it. You just hope they honor it. Most will.
Yes, complete Hollywood trope.
AIUI, you’re expected to reach an agreement with the reporter before the interview as to whether your answers will be on the record. Randomly prefacing a remark with “off the record” during an interview is just asking for trouble. (And they really don’t like “off the record, no comment.”)
My entire career covered three components: regular Army, Army Reserves (drilling), and regular Navy. Not even once did I see that done.
<< removed by me…not FQ stuff >>
I never asked for permission. I simply spoke my mind.
Did you ever hold your tongue because what you had to say might not be well received?
With some regret, not that I remember.
I can also add that in nearly twenty years as a civilian working for the Navy (alongside many active duty Navy personnel) I’ve never seen this.
I once had to ask for permission to speak. Not speak freely, just speak at all. The other person, my immediate supervisor at the time, kept interrupting me when I tried to explain why I’d done something that was not quite the way we usually did things. Finally, I came to attention and said “Permission to speak.” This was in the Air Guard, so in the military. Not something that happened again.
There you go again. You just did. 
In my experience, outside of very specific formal or technical situations, military people talk exactly like normal human beings.
Enlisted have to call officers “Sir” at all times, don’t they? I don’t think most civilians ever use the word sir towards their boss.
The female officers I worked with wouldn’t have appreciated that. 
But “Sir” or “Maam”, or rank. Even with commissioned colleagues who I considered personal friends, I never got on a first-name basis.
But they’re probably used to the occasional slip up.
You throw it in there from time to time usually at the beginning of a conversation. You don’t have to use it in every sentence. When you have a close working relationship like platoon leader/platoon sergeant it’s usually less formal, unless the lieutenant is a dick.
Asking permission to speak freely is ok as long as you follow it with “With all due respect.”
Make sure to toss in a smattering of, “No offense intended…” and, “I’m just saying..”
During basic I had two women instructors, one demanded to be called sir and the other ma’am. But that’s the mind games they like to play on young trainees. If you slipped up, you got chewed out. Once out of basic however, almost all formalities drop. During my time only met a single E-5 who demanded all lower ranks stood at attention whenever he entered a room. Additionally all officers I spoke to casually said that mandatory saluting was a drag.
As too the OP I can only speak from personal experience, one time whilst getting reprimanded by an 0-6 (for being late to a staff meeting) he granted me permission to speak freely while at attention. I thought this a trap and basically said nothing.