Why Fighters?
When anybody thinks "space fighter," they are analogizing to fighter aircraft. After all, space is kind of like the air, just higher. And even the Air Force handles the military's space stuff! This must mean space fighters make sense, right?
Wrong
Let's look at a couple of existing types of warfare:
- Ground warfare
- Air warfare
- Naval surface warfare
- Submarine warfare
- Space warfare
All of them take place in different environments. None are equivalent to any other; in particular, space combat is not the same as naval combat or submarine combat, or air combat.
Before we try to generalize the concept of a fighter to space, which people have much less intuition about, let's try to generalize it to the other military branches. I'm going to define a fighter as a small one- or two-man vehicle, that places emphasis in combat on speed, maneuverability, and positioning.
- Land fighter: The closest thing is probably a tank or Humvee. However, neither tend to shoot on the run: they place more emphasis on protecting their occupants with armor. Tank combat is more like hide-and-seek than tag, unlike aerial dogfights.
- Sea fighter: As far as I know, there are no one-man ships. As far as I know, very small boats are typically used to move people from ship to shore and are not pitted against one another on the battlefield.
- Submarine fighter: The smallest combat submarines have dozens of people onboard. Small submarines are used only as research vessels, and always operate with a 'mothership.'
Space is Hard
If the fighter concept doesn't work for land or sea, it seems unlikely to work for space either. However, there are a few points which turn that into a definitely.
- Humans are heavy. The smallest possible human-carrying spacecraft would be something like Mercury: 6 feet wide by 10 feet long (2 x 3 m) and weighing only 1.5 tons, the only maneuver it could perform was a single reentry burn. It required a 30 ton launch vehicle, which barely lifted it into orbit. Also note the fact that this spacecraft was not reusable, and had only enough life support equipment for around a day (reusability and longevity add significant weight, just look at Shuttle). The military typically likes their aircraft to last multiple decades, at least.
- Humans are squishy. Even trained fighter pilots won't last long past 10 g's of acceleration. Sounding rockets can easily hit over 20 g's during launch, riding one would be potentially lethal to a human. Even heavy-lift vehicles ramp up from around 1 g to over 5 gs over the course of four or five minutes. The long and continuous application of g-force could also easily be fatal.
- Humans are bad at space. As was pointed out, orbital mechanics are unintuitive. Although this could be mitigated to an extent with training, there's no getting around the fact that maneuvers in space require timing and control far beyond what a human could achieve. The other problem is that piloting a real-life space fighter would involve hours of tedious scanning the sky with a telescope for targets, then recording orbits and calculating trajectories, culminating in a rendezvous at 10 to 15 km/s. Humans don't work slow enough or fast enough to handle either part well. Play around with Orbiter for a bit to get a feel for the problem.
We've actually already solved most of these problems: we send computers into space! They can be extremely small, with low power requirements, and can handle extreme conditions of temperature, vacuum, and acceleration.
Since there are no assets to capture in space (no resources [that would be economical to obtain] or cities [see, space is a bad place for people, above]), space combat will likely be all about destroying your opponent's spaceborne assets (weapons platforms and intelligence-gatherers). For this task, only one form factor makes sense: an anti-satellite missile.
So to answer your question... (tl;dr)
The closest equivalent to a space fighter would probably be slung under a real fighter, carry no people, and spend less than five minutes in space before exploding. Hey, you asked for realistic, and reality is disappointing. Sorry.
(Realistic spaceflight is only fun for hardcore space nerds like me, handwave away the problems and let yourself have fun with it instead!)
Update: Star Wars
celtschk makes a good point: if space fighters don't make sense for offense, maybe they make sense for defense? To answer this, let's take a trip back in time to the 1980s; no, not for movies, for the other Star Wars.
At this point in history, something like antisatellite weapons already exists, called ballistic missiles. Antisatellite missiles existed too, but ballistic missiles were the bigger threat. Lots of people were figuring out ways of stopping said missiles.
Typically you can't stop the missiles in their boost phase, before they leave the atmosphere, since this would typically entail having weapons in your enemy's airspace, which they don't really like. It's also hard to stop the warheads when they reenter, since they're moving very fast. Therefore the only place to stop them is in space.
Since explosions don't work in space, you'd stop a warhead (or antisatellite missile) the same way that an antisatellite missile would destroy a satellite: just before impact, explode into a cloud of shrapnel that strikes the target, ripping it to shreds. There is no defense against this, especially when the closing velocities are tens of kilometers per second.
Here, the same disadvantages for humans that I mentioned above are even stronger. You want your weapons to be extremely small and light so that they can be put on an intersecting orbit within seconds. If Star Wars systems were ever deployed, they would have been completely automatic, as having a human in the loop makes the response too slow to be effective. You also need your interceptors to be inhumanly accurate and expendable (see, "explodes into shrapnel," above).
A plausible defense system consists of IR and UV cameras aimed at the Earth (both detect heat from the exhaust plumes of missiles, but UV has the advantage of not penetrating the ozone layer, so you only see spaceborne sources), and a constellation of kinetic kill vehicles that coordinate to intercept and destroy each of the incoming warheads. These systems were called Brilliant Eyes and Brilliant Pebbles, respectively.
Why are these systems not in place today? You run into a scaling problem. Your enemy can release not one, but dozens of warheads from a single missile. All of them but one are inert. Since they are basically cans, they cost almost nothing to add to your missiles. However, you can't distinguish which ones are which, so you have to deploy enough interceptors to destroy all of them. The catch is, none of your interceptors can be duds! They all have to be fully-functional, expensive systems.
This is why laser systems that destroy missiles in their boost phase before they can separate their warheads are so attractive. However the same issues of detection, targeting, and speed come into play, so any laser weapon would be computer-controlled. You might stick a person on board to deactivate the system in case of a false alarm, but then you need all the support equipment for that person, making you a huge target.
Beyond The Infinite
There is one change that could make space fighters plausible: reactionless drive. All spaceflight under known physics is fundamentally limited by the rocket equation:
$$
\frac{m_f}{m_p} = e^{\Delta v/v_e} - 1
$$
This equation comes from the fact that momentum is conserved, so in order to change your momentum, you have to emit something with momentum opposite the change you want to make. The reason this equation is so bad is that the change in speed is in the exponent, meaning that the required fuel mass fraction increases dramatically as you increase the amount of maneuvers you can make. But what if:
$$
\frac{m_f}{m_p} = 0?
$$
A reactionless drive doesn't require any propellant to be expended. This would change the whole game, as now spaceflight is no longer a mass-minimization exercise. You can put anything in space that you want, and move around as much as you want once you're up there. This invalidates most of my arguments against space fighters, although it requires breaking the laws of physics to do it!