What is known about the possibility of a "real King Arthur"?

I'm a little surprised that no one has mentioned, among the various theories so far, the possibility of the figure of King Arthur being (at least loosely) based upon the military leader Riothamus. Whether or not Arthur was real and the stories about him true or embellished, there are - as another answerer said - "plenty of theories regarding the existence of such a person". At least one worth mentioning, which I first learned about from watching a documentary on the subject, is that Arthur was Riothamus.

I will leave the following quote from Wikipedia to speak for itself:

Riothamus has been identified as a candidate for the historical King Arthur by several scholars over the centuries, notably in recent times the popular historian Geoffrey Ashe. Whether or not this identification is true, Riothamus' activities in Gaul may be the seed whence grew the tradition (first recorded by Geoffrey of Monmouth in his Historia Regum Britanniae) that Arthur crossed the English Channel from Britain and attacked Rome. Geoffrey Ashe has also suggested a link between Riothamus' alleged betrayal by Arvandus and Arthur's betrayal by Mordred in the Historia Regum Britanniae, and notes that Riothamus' last known position was near the Burgundian town of Avallon, which he suggests is the basis for the Arthurian connection to Avalon.

The academic historian Léon Fleuriot argued that Riothamus is identical to Ambrosius Aurelianus, an historical figure in Britain around this time who, in the early narratives containing Arthur, preceded Arthur. Fleuriot suggested that "Riothamus" was Aurelianus' title as overlord of all Brythonic territories. He noted that "Riothamus" and Aurelianus are contemporaneous and that Aurelianus is the only British leader of the time who is identified (much later) as ruling both Brythons and Franks, which could only be the case if he ruled territory in Brittany. He also suggested that the name "Abros" in Breton genealogies is a contraction of "Ambrosius" and that Nennius indicates that Ambrosius was supreme ruler of the Britons, which, Fleuriot argues, would translate as "Riothamus". Fleuriot suggested that Ambrosius led the Britons in the battle against the Goths, but then returned to Britain to continue the war against the Saxons.

Source: https://en.wikipedia.org/wiki/Riothamus

What is certainly known about the possibility of a real King Arthur . . . is that it's possible he existed. That much is noncontroversial.

However, I'm of the "where there's smoke, there's fire" school of thought, meaning I believe the "possible" can be pushed to become "probable." Here's the skeleton of my argument.

1. The Battle of Badon is generally viewed as a real historical event. Not only is it mentioned in a nearly contemporary source (Gildas), but there's also confirming historical and archeological evidence demonstrating a temporary halting of the Saxons' westward incursion, for a couple of generations. This interruption coincides with the documentary timing of Badon.

2. Someone was no doubt the overall commander of the Brythonic forces. This person would be our theoretical "Arthur."

3. The only name ever explicitly given, by any writer, to this commander is "Arthur." No competing name has ever been put forward by any ancient source - unless we take Gildas's mention of Ambrosius Aurelianus, several sentences before the mention of Badon, as indicating the name of the commander. However, in the immediate context Ambrosius can be understood as instigating the Brythonic resistance to the Saxons one or two generations before Badon, rather than necessarily having been the one who personally led the troops at Badon a generation or two later.

On balance, these considerations incline me to believe Arthur really existed - and by that very name; it wasn't merely a title or nickname. However, since in the immediate several generations following Badon there were several "Arthurs," all of them Irish, and since there were multiple Irish settlements on the island both before and after Badon, I believe (for now!) that Arthur was of Irish stock (though perhaps also partly Brythonic). This would be consistent not only with the "Irishness" of his name, but also with the Historia Brittonum's distinction between Arthur and "all the kings and military force of Britain . . . . And though there were many more noble than himself, yet he was twelve times chosen their commander": i.e., Arthur was not himself a Brythonic king, nor did he match their tribal/political nobility.

In this connection, an intriguing little piece of relevant history was uncovered at the Roman and subRoman Viroconium (adjacent to the village of Wroxeter), sometimes argued to be "Camelot" due to its early-6th-century rebuilding under a Brythonic (or Irish???) warlord. At that time it was the island's 4th-largest city. "A tombstone found [at Viroconium] in 1967 in ploughing bears an inscription to Cunorix. The use of the word macvs for “son of” is an Irish form and dates the stone to the late 5th c. or later. Probably the man was an Irish mercenary employed by the citizens to protect them from wandering bands of brigands." (Richard Stillwell, et al, eds., "VIROCONIUM CORNOVIORUM (Wroxeter) Shropshire, England," The Princeton Encyclopedia of Classical Sites, http://bit.ly/290wEKY [emphasis mine])

The single-best argument in favour of Arthur's existence that I've yet seen is Christopher Gidlow's The Reign of Arthur (The History Press, 2007). I can't recommend it highly enough.

There is no historical or archaelogical evidence to support the existence of a King Arthur.

What there is, are plenty of theories regarding the existence of such a person.

Your last sentence is probably (in my opinion at least) the closest to the truth.

There was or may have been a 'legendary' warrior existent at that time whose prowess, bravery and military strategy ensured his success on the battlefield whilst fighting the invading saxons became the stuff of legend and folklore.

There's plenty of historical anecdotal evidence that supports the view that the people of that time were highly superstitious and held great belief in 'omens' and 'signs' from the gods that they may well have elevated a successful warrior to kingly or evenly god like status.

Based on a factual person, this kind of character could quite easily have morphed and evolved over the many years since it's happening into the legend we like to believe in today.

Given that Arthur comes from the latin name 'Artorious' and that the Romans themselves were renowned for their military prowess, it's highly likely that the figure we know of as Arthur had some connection to the Romans, and was probably skilled or trained in their fighting methods.

Bernard Cornwell wrote an excellent trilogy of books on the King Arthur legend and I would highly recommend them as fascinating reading for anyone interested in understanding who King Arthur may have been.