Can anyone shed some light on this? All I can think of is that it has something to do with the chromosomes being paired
It is still considered a Eukaryotic cell because the daughter cells and mother cell are both Eukaryotic; the chromosomes will condense and be contained in a Nucleus after Telophase and Cytokinesis. Furthermore, Prokaryotic binary fission does not involve the assembly of Spindle Fibers, whereas dividing Eukaryotic cells in Anaphase use Spindle fibers to transport the Chromosomes towards the poles of the dividing cell.
I hope this answers your question!! :)
Only eukaryotic cells are able to under go a mitotic/meiotic cell cycle, in which, after an interphase of variable time length, such cells enter M-phase, and one of the regarded phases of such stage is anaphase, so in order for a cell to undergo chromosome segregation in the form of anaphase, it has to be eukaryotic.