In the "real world" people in a relationship shouldn't be in the same unit for any number of reasons, especially if one is a superior. For example - if you're in command, you might need to order someone to their death. Could you do that with someone you're in love with?
That doesn't even get into issues like performance evaluations, promotions, etc, that I'm sure commanding officers are responsible for. Many civilian businesses and government agencies don't allow managers to get involved with their employees for that reason, so it's not just the military. On Capitol Hill, these policies vary by office. However, my office does NOT allow relationships between someone in a management position and a staffer they oversee. If such a relationship comes to light, the employee handbook clearly states that the relationship must end, or one staffer must either leave the employment of the Senator entirely or leave the D.C. office by transferring to an "in-state" office (if an opening is available at one).
As for my post-war head canon:
1. Shepard spends just over a year recovering after being badly injured. Ash takes a leave of absence, and they spend a significant amount of that time together. Shepard is holed up in the cabin he got in the Pinnacle Station DLC, so everything is relatively out of sight.
2. The Normandy was pretty seriously damaged during the final battle, so it's in dry dock on the Citadel. Shepard acts as a consultant for (yet another) retrofit during the repairs. There aren't many changes, but one of the observation rooms is turned into a lab - he always did think it was stupid that they got rid of the original lab, which he thought came in handy. The "war room" remains unchanged - the post war goal of the Normandy will be to gather intelligence and "put out emerging fires" in the chaotic post-war galaxy before they cause too many problems.
3. The Alliance discovers that the Normandy SR-2 needs an AI to operate at peak efficiency. They determine that EDI can be repaired, but they need to bring the Council and the salarians in to help. As part of the deal for allowing an unshackled AI on an Alliance ship, the Alliance has to agree that a Spectre must be captain.
4. After his recovery, Shepard is promoted to Captain and given command of the Normandy. Several old crew members (including Ash) come back, along with a few new ones. Along with a mostly Alliance crew, intelligence officers from the Council powers are also assigned to the ship, the idea being that pooling and coordinating intelligence efforts is the best way for the Normandy to achieve its goal of promoting post-war stability. The ship has a new STG scientist manning the lab, albeit one that's actually not completely fond of Shepard - he disagrees with curing the genophage and thinks Shepard has "simplistic" views of the galaxy.
5. Despite their efforts to be discreet, the crew of the Normandy figures out pretty quickly that Ash and Shepard are together, but choose to ignore it. It never makes its way back to the Alliance brass, and given the circumstances behind EDI's reactivation, there's not much the Alliance could do about it anyway.
6. The Normandy is successful at putting out several fires, but the galaxy as a whole is still a mess. The Alliance is suffering from poverty and poor living conditions on Earth, conditions that are allowing some political extremists to gain traction. Councilor Osoba is assassinated. The salarians and what's left of the batarians eventually both end up in full scale civil war. There's an attempt to infect the krogan with a new genophage. There are several attempts on Wrex's life, and eventually even a conflict there between the factions behind Wrex and the most recalcitrant traditionalists. The geth were so devastated by the firing of the Crucible that they were thought to be wiped out - until the Normandy discovered otherwise. The geth that remain are split between a faction that wants to stay isolated while they figure out their new "individuality," and another that decided it was a mistake to back organics in the war and wants revenge for what the Crucible did.
7. Shepard commands the Normandy for 3-4 years, then leaves to take a new position overseeing Spectre operations and advising the Council (the position is meant to help address some of the problems that have popped up with Spectres, an issue that is on going even after the war). He's promoted to Admiral, and remains nominally connected to the Alliance. Ash gets command of the Normandy. Shepard goes out into the field now and then, but mostly he's in a desk job at this point.
8. Ash eventually takes a Citadel based position that allows them more time together. She goes out into the field more often than Shepard, but not as much as in the past. They eventually get married, have kids.
9. Shepard reluctantly takes the human slot on the Council - he has been playing a role in trying to end the salarian civil war, which has been dragging on without resolution, and the existing Council members felt that he would be better positioned to address this conflict (and others) on the Council. To his surprise, he finds he's pretty good at it, and stays on far longer than he thought he would - though there are times he thinks its fitting punishment for putting Anderson into the role. (NOTE: Being appointed to the Council requires him to temporarily surrender his Spectre status, as it's felt that having a Spectre on the Council would raise too many potential issues).