What is the possibility of Roman rump states, former Roman provinces, or nation's based partially on Roman culture being created or surviving in Western Europe during the decline of the Western Roman Empire?
Well, then Barbarian Kingdoms really fit what you describe.
See, Barbarians didn't so much invaded Romania, than progressivly replaced the roman imperium with their own : Barbarian institutions in the Vth century were largely a continuation of late Roman institutions on virtually all matters. It was helped by the fact various Barbarians, as peoples, didn't appeared before the IIIrd century, and did so trough Roman policies and influence.
Goths, for exemple, were a mix of Dacians, Sarmatians, Germans and Romans structurating as chiefdoms in the IIIrd century, trough interaction with Romania : trade, military recruitment, raids, etc. All of this that made Goths a romanized people right from the beggining. As Barbarian peoples entered in the Empire, they were even more integrated within late imperial structures, going trough an even greater demographical mix-up and romanization.
By the Vth century, making the difference between a Barbarian and a provincial Roman isn't easy, as the Barbarian identity was essentially political. : you were Barbarian because you followed your Barbarian king, you were Roman because you served the Roman state (as a German-issued Stilicho did). Of course, this model is mostly theoritical, and you had more than just than playing (Stilicho was seen at best as a semi-Barbarian, after all), but it ratger fit.
It changed with the Vth and the disappearence of the Roman state in the West : it lead Romans to directly serve who had the imperium beggining a quick political mix (being understood that Barbarian peoples were importantly romanized and gathering Roman-issued people in first place).
Simply said, Barbarian kingdoms were in virtually all matter a continuation of late imperial institutions (political or administrative) and culture (most, if not all, Germanic specches used by Barbarians settled in Late Romania disappeared in the VIth).
We can still distinguish in the Vth, regions with
- A clear continuation of late imperial romanity in ERE, Africa and Italy and parts of eastern Spain and southern Gaul which maintained for a while the municipal institutions in cities and the late classical spectables and schools on a large scale (it did existed elsewhere, more not distintively)
- A post-imperial culture, marked by episcopalian institution in cities, a clerical scholarly culture, etc. It mostly concerned Gaul, Spain, Mauretania, etc., even if post-Ostrogothic Italy looked like this more and more.
Regardless of this relatively blurry distinction, you didn't have that much changes in law and customs, political frames, institutions, military organisation, etc. from late Roman culture. It did evolved from there, of course, but the key words are "evolved from there", not disappearing.
Now, there's the case of peripherical regions such as Illyricum, Moesia and critically Britain, on which late imperial institutions and features were significantly less present. It's less due to the political takeover of Barbarians, tough, than these regions being undepopulated and underdevelloped structurally compared to the majority of Roman provinces : while peoples as Franks evolved within the frame of old provinces and diocese, Anglo-Saxons exploded into tribal kingdoms because you didn't have this much presence of Roman structures and influence to begin with before the Vth.
Except the British situation, while you had a Germanic influence in post-imperial Romania, not only it tended to be really superficial (contrary to what
@Sceonn argues), but it could be even posterior to the creation of Barbarian kingdoms : most of Barbarian laws (which are essentially inspired by Roman law) were written in the Vth/VIth centuries, "identitarian" objects as fransiscae, clothes or eagle fibulae simply don't exist before the Vth and VIth, etc.
I know that their was the
Kingdom of Soissons, a rump state in northern Gaul ruled by
Aegidius, although I am not sure if it was fully de-facto independent.
The problem isn't much to know if it was independent or not, but rather to know if it was real or not in first place.
See, the whole idea of a Gallo-Roman entity stretching from Brittany to Rhine in opposition to Franks comes from the XIXth century, when historical carthography apparead, and when there was a "gap" of knowledge of Northern Gaul.
What do we have, in ancient sources, about Syagrius? Grégoire of Tours, which mentions him twice.
Ægidius died, letting a son named Syagrius
And, much later.
In the fifth year of his reign, Syagrius, king of Romans and son of Aegidius, was in the city of Soissons, whom Aegidius took once
Syragrius (from the famous Late Antiquity and Early Medieval Gallo-Roman family of Syagrii) as some local Gallo/Hispano-Roman rulers was probably more or less independent, but essentially locally so : in clear, Syagrius may have ruled only around Soissons, or at best up to Seine, with blurry borders with Franks (it's even possible that Soissons was held at some point by Chilperic), but nothing like an unified demesne betwen Somme and Loire. We know of at least one other comes in the region, the Franko-Roman Arbogast, and you probably had more that weren't named by Grégoire.
It's interesting, then, to see that Clovis didn't took control of Orléans in the wake of Syagrius' defeat, but a decade later : Syagrius' power might simply have been reduced to a region including, roughly, Soissons, Noyon and Senlis; the rest of northern Gaul being a shifty ensemble of
Saxons, Alans, Franks (Bray and Caux regions), and other more or less romanized Barbarians together with Gallo-Romans (and possibly with
Ambrosius Aurelianus/Riothamus as well)
It's likely that it was essentially the emanation of the Gallo-Roman land owners, supported by private armies (bucellarii) probably as much Barbarised than the Frankish forces (which were, for all intent and purposes, the Roman army for Lower Rhine, and acknowledged as such), without a clear leadership.
The institutional and cultural difference between Syagrius (whom a kindred wrote the Burgundian Law) and Clovis were probably minimal, the latter having probably more legitimacy and more cohesive power than the former.
As said, Syagrius wasn't really unique, as a local remnant of provincial power : there's Vicentius in Taracconensis, Apollinaris Sidonius and Ecdicius in Auvergne, Victorius, Desiderius and Namatius* in Aquitaine, Syagrius and Avitus in Provence, Arbogast in Germania, etc.
These men were invested (or, more often than not, invested themselves) in the late Empire with administrative (
militia) roles would it be military (Vicentius was probably trusted the military charge of Taraconnensis by Majorian) or civil (which was generally translated by an episcopalian position, but not always).
These men and their lineages eventually joined up Barbarians, as part of their kingoms' militiae, because the fall of the Roman state in West only let Barbarian imperium as a legitim authority coming from imperial institutions. Similarily, Barbarians rulers readily accepted them as part of their military and administrations, because not only it strengthened their own power (the death of Vicentius was a blunder for Euric's power in Spain, for exemple), but it strenghtened their role as successors of the empire.
*Namatius was possibly related to two others Namatiu one bing bishop of Orléans, and the other bishop of Vienna, both in the VIth century.