There are overt anarchist and communal themes in François Rabelais' seminal work Gargantua, originally published as La vie très horrifique du grand Gargantua, père de Pantagruel in Paris, 1534. In the work's concluding chapters, following a massive (to the point of comic absurdity) battle, the Abbey Theleme is founded by the sole surviving monk, who happens to posses an early modern "punk" sensibility (punk + monk =...munk? Ponk?)

Gargantua, Chapter LII (trans. Thomas Urqhart emphasis mine)

There was left only the monk to provide for, whom Gargantua would have made Abbot of Seville, but he refused it. He would have given him the Abbey of Bourgueil, or of Sanct Florent, which was better, or both, if it pleased him; but the monk gave him a very peremptory answer, that he would never take upon him the charge nor government of monks. For how shall I be able, said he, to rule over others, that have not full power and command of myself? If you think I have done you, or may hereafter do any acceptable service, give me leave to found an abbey after my own mind and fancy. The motion pleased Gargantua very well, who thereupon offered him all the country of Theleme by the river of Loire till within two leagues of the great forest of Port-Huaulx.

He goes on to adorn the abbey's gates with the proto-Anarchist/Situationist "Do What Thou Wilt"

Ibid, Chapter LVII

All their life was spent not in laws, statutes, or rules, but according to their own free will and pleasure. They rose out of their beds when they thought good; they did eat, drink, labour, sleep, when they had a mind to it and were disposed for it. None did awake them, none did offer to constrain them to eat, drink, nor to do any other thing; for so had Gargantua established it. In all their rule and strictest tie of their order there was but this one clause to be observed,

Do What Thou Wilt;

Although François Rabelais writes in the 16th century, at the very end of what one might construe as the 'Middle Ages', his audience was the doctoral ecclesiastical class whose upbringing and education is rooted in the medieval Scholastic tradition.

But Rabelais, brilliant as he may be, is speaking of mere hypotheticals. For a more 'realistic' view into 'medieval anarchism', we can turn to 1353, when Boccaccio wrote a most fascinating journalistic Introduction to The Decameron. While more first-hand documentation than any sort of actual critical framework of 'anarchism', he describes the conditions of his city of Florence during the Plague, examining the political breakdown and communal restructuring in its wake

The Decameron, Introduction (trans. Richard Hooker, emphasis mine)

Others, who disagreed with this, affirmed that drinking beer, enjoying oneself, and going around singing and ruckus-raising and satisfying all one's appetites whenever possible and laughing at the whole bloody thing was the best medicine; and these people put into practice what they heartily advised to others: day and night, going from tavern to tavern, drinking without moderation or measure, and many times going from house to house drinking up a storm and only listening to and talking about pleasing things. These parties were easy to find because everyone behaved as if they were going to die soon, so they cared nothing about themselves nor their belongings; as a result, most houses became common property, and any stranger passing by could enter and use the house as if he were its master. But for all their bestial living, these people always ran away from the sick. With so much affliction and misery, all reverence for the laws, both of God and of man, fell apart and dissolved, because the ministers and executors of the laws were either dead or ill like everyone else, or were left with so few officials that they were unable to do their duties; as a result, everyone was free to do whatever they pleased.

One final example I can think of is perhaps the most interesting and also the most mysterious. In Medieval Anatolia, groups of young tradesmen sprung up in cities and formed brotherhoods based around the moral precepts of Futuwwa, a sort of chivalrous ideal for young Muslim men that emphasized selflessness, kindness to strangers and communal unionized spirit. Ibn Battuta, famed traveler and 'anthropologist', writes of his experiences with them in 14th Century Turkey

The Travels of Ibn Battuta, 'The guild organisations of Anatolia, their leaders, the Akhis, and their members, the fityan' (trans. Beckingham, Gibb, emphasis mine)

Nowhere in the world are there to be found any to compare with them in solicitude for strangers, and in ardour to serve food and satisfy wants, to restrain the hands of the tyrannous, and to kill the agents of police and those ruffians who join with them. An Akhi, in their idiom, is a man whom the assembled members of his trade, together with others of the young unmarried men and those who have adopted the celibate life, choose to be their leader. That is [what is called] al-futuwwa also. The Akhi builds a hospice and furnishes it with rugs, lamps, and what other equipment it requires. His associates work during the day to gain their livelihood, and after the afternoon prayer they bring him their collective earnings; with this they buy fruit, food, and the other things needed for consumption in the hospice. If, during that day, a traveller alights at the town, they give him lodging with them; what they have purchased serves for their hospitality to him and he remains with them until his departure. If no newcomer arrives, they assemble themselves to partake of the food, and after eating they sing and dance. On the morrow they disperse to their occupations, and after the afternoon prayer they bring their collective earnings to their leader. The members are called fityan, and their leader, as we have said, is the Akhi.

Battuta's writings on the Young Akhis, which comprise only two or three chapters, are probably the most substantial sources we have regarding these organizations so it is really an open question how 'anarchist' or communal they truly were. However there is no doubt they pursued a notably coöperative, DIY lifestyle at the very height of the so-called 'Middle Ages'.

I think the Akhis are the closest to what you are imagining. But really it comes down to a question of terminology and perhaps imagination. Is a leaderless free for all, in the case of Boccaccio's post-Plague Florence, considered 'Anarchism' in the political sense that our Monty Python peasant is using it? Are the Young Akhis proto-syndicalists, religious zealots, or somewhere in between, distant kin to Tolstoy's Christian Anarchism? Is "anarchism" a young clergyman, desperate to escape the rigid hierarchy of the School and Church, duskdreaming during the vespers of a Theleme they could call their own?