According to Orderic Vitalis and a 13th century chronicle from Laon known as the Chronicon Laudunense, a large group of Anglo-Saxon political refugees fled England in the 1070s on a fleet from England to Constantinople and a stream of immigration continued in the decades following as a result of the dispossession of the Anglo-Saxon nobility and their households. Orderic Vitalis, The Ecclesiastical History ed. Chibnall (Oxford: 1969), Book IV, 202.

According to Krijna Nelly Ciggaar, their decision to leave for the Empire in particular (in addition to places like Ireland and Scandinavia) was motivated by the fact that it was commonly known in England, and indeed throughout the Latin Christian West, that the Byzantines were actively recruiting mercenaries from England, Northern Europe and Scandinavia. Krijna Nelly Ciggaar, Western Travellers to Constantinople: The West and Byzantium, 962-1204 (Leiden: Brill, 1996), 140-141. There would almost certainly be a significant demand for your services if you were a capable warrior during this period and you would likely find opportunities to serve in some capacity. Whether or not you would make it into the ranks of the Varangian guard is a different matter, and would be informed by your birth and rank.

Ciggaar states that certain important positions such as bodyguards, palace guards and keepers of the treasury were reserved for men from these regions. It appears that Anglo-Saxons did enter the ranks of the Varangians quite quickly and are recorded fighting in that unit at Durazzo in Geoffrey of Malaterra's history in 1081. De rebus gestis Rogerii et Roberti Guiscadi ed. E. Pontieri, 74.

Anglo-Saxon names are listed among the ranks of the Varangians all the way into the early thirteenth century and are recorded by Robert of Clari in his account of 1204 suggesting a pattern of longterm settlement and successful service that lasted up to the Fourth Crusade.