In historical linguistics, when a word or string of words is reinterpreted as consisting of a different set of words (for example, when "an ewte", which is what people used to say in Middle English, becomes "a newt"), they call it reanalysis. Here are two somewhat parallel examples.
In classical Arabic, one word for "he came" is jā'a. "With" is bi-. "He came with X" is jā'a bi-X, and can usually be translated as "he brought X". In some parts of the paradigm, the two words remain more or less adjacent* - eg ya-jī'u bi- "he comes with"; in others, they are separated by an agreement morpheme - eg jā'-at bi- "she came with", ji'-nā bi- "we came with". In all modern dialects, the glottal stop is lost, and so are the final short vowels, which would regularly yield jā b(i)-, yijī b(i)-, jā-t b(i)-, etc. But in fact, this common construction was reanalysed as a single word, so you get forms along the lines of jāb, yijīb, jāb-it, jib-nā...
In Proto-Berber, as across most Berber languages, the word for "come" was something like as (perfect form y-usa, habitual yə-ttas, etc.) However, Proto-Berber also had a very productive system of "extensions", particles near the verb marking the direction in which the verb's action took place: towards (d) or away from (n) the speaker. Naturally, "come" normally featured the d extension. In many common forms, it was adjacent to the stem (eg y-usa d "he came", nə-ttas d "we come", etc.); in others, it was not (eg ad-d as-əγ "I will come", usa-n d "they came", etc.) In at least one variety - the dialect of the Beni Snous near Tlemcen, in western Algeria - this d was reinterpreted as part of the word "come"; so there (with voicing assimilation of s to z when next to d) you get forms like yusəd, nəttasəd, ad azd-əγ, uzd-ən.
* Strictly speaking, even in this one they're separated by a short vowel marking mood.