In the TNG episode "The Wounded" Benjamin Maxwell says that Miles O'Brien served as his tactical officer on the Rutledge.
However, I had thought that O'Brien was enlisted, not an officer. How could he serve as tactical officer (a bridge officer) if he was not commissioned?
Out-of-universe, O'Brien's rank and status as an enlisted man was inconsistent early in TNG. But in-universe, Starfleet captains appear to have the ability to name non-officers acting officers. Most notably, Wesley Crusher is named an acting ensign and given a bridge officer position despite not even being in Starfleet.
PICARD: Commander Riker has told me how supportive you were in Engineering. Well done. At ease. Sit here in Command.
RIKER: Captain, it's not allowed. Your orders.
PICARD: Oh, that's true. Well, I can't waive them again. Only commissioned officers.
WESLEY: It's quite all right, sir, I understand.
PICARD: Please don't interrupt me, Wesley.
WESLEY: I'm sorry, sir.
PICARD: Any commissioned rank? Even ensign?
RIKER: That would give him authorised access to the Bridge.
PICARD: Well, then, I'll have to make him an acting ensign. Captain's log, stardate 41263.4. For outstanding performance in the best of Starfleet tradition, Wesley Crusher is made Acting Ensign, with the duties and privileges of that rank. And whether that rank becomes permanent, Mister Crusher, depends on you. At the earliest opportunity, your entrance application for Starfleet Academy will be tendered. Until then, you will learn this ship. Every operation, every function. Commander Riker, a duty schedule for Mister Crusher, heavy on study. (Where No One Has Gone Before)
Picard later gave Wesley a field commission, promoting him from acting ensign to full ensign.
PICARD: The Academy must make you wait, that's true. But, when I review your service to this ship, your crewmates, I cannot in all conscience make you wait for the Academy. You see, Wesley, in my eyes you're an acting ensign in title only. I hereby grant you field promotion to full Ensign, with all the commensurate responsibilities and privileges of that rank. Congratulations. You're dismissed. (Ménage à Troi)
Likewise, Star Trek 2009 features the bulk of the cast serving as acting bridge officers even though none have graduated Starfleet or earned a commission, while Voyager sees Chakotay and Paris receive temporary commissions despite being kicked out of Starfleet entirely.
If a teenage boy can be named a full ensign in peacetime at Picard's discretion, it's not a stretch to imagine that O'Brien was given a battlefield promotion to a bridge position at the height of the Cardassian War.
O'Brien is an officer, he's just not commissioned. He's worked hard and has earnt his ranks up from the bottom.
It seems from TenthJustice's answer and the quote therein that the 'Commissioned' officer rule is Picard's rule and not Starfleet... therefore it is likely that it is generally down to the Captain who serves on the Bridge.
O'Brien also serves as the Operations Chief on DS9, which is a position held by Data on the Enterprise... Data is commissioned.
It would make sense that in the inclusive nature of the Federation and Starfleet that roles and titles are based on the person's ability and not how they entered Starfleet.
As a 'real world' example, it was common in the late 1700's and Early 1800's for suitably experienced NCO's within the British Royal Navy (petty officers for example) to be given temporary command of a prize of war, should the vessel not warrant someone more senior, or the attacking vessel not having a suitably ranked individual.
Once the Captain has made up his mind the discussion is over.. – Edmund Dantes Jan 11 at 8:17