He suggests that he needs to "die" in order to be in his ghosty form. So Relius forces his spirit out of his body by killing him (hence why he's in ghost form when he attacks the takamagahara). I think it would've happened anyway if Hazama had just bled out but it was basically along the lines of "If you need to die anyway, why don't we duke it out."
If you fight Relius vs. Hazama (any game mode), Relius says "You waste our time with these antics." which I believe is partly a reference to this bit in the story.
In the story mode, immediately after Relius kills Hazama, Hazama does the "put a virus in Takamagahara" scene.