The best existing candidates have both been aired. As I said in my comments on Oerlekens, one problem is that ‘the Greeks did not have a word for it’. Indeed, they did not have punctuation at all. Failing the Greeks, the Americans are very good at coining words and phrases. So ‘run-on’ is in origin an American English expression.
But today’s Greeks do have a word for punctuation which might do: stixi (στίξη). It is derived from the ancient word stichos (στιχος), meaning a rank of soldiers. But the word came also to denote musical notation. From that it must have been adopted as the best word for punctuation.
From there, we might be able to construct the literary term Matthew Sbarr is seeking.
To turn it into its negative, you would add the *privative alpha, as in astixia. It might be astixis; or it might be astichism. Either would be a legitimate coinage. Because neither has been used before, so far as I can discover, you have a free choice. I prefer the former.
The trouble with this, however, is that before it can be used, you or someone has to publish an article on the subject and hope some readers run up the flag and salute!
Short of that you are left with ‘run-on’ and ‘stream of consciousness’, neither of which is ideal.
As a last resort, I wonder if ‘the Germans have a word for it’: they often do! In fact, looking up the word unpunctuatedness, I find the word unpunktiertheit.
This, in turn, has led me to discover that critical commentators on Joyce do use the term unpunctuatedness to express the quality of some of his writing. For example, Derek Atridge uses it in Joyce Effects. This refers to the quality of the practice rather than to the practice itself. But there it is.