Probably a better way to describe this, but I think this is a good start:
Some multifunctions have analytic sheets outside a branch cut, others do not. The function does and we can show this by considering an arbitrary point off the branch cut and writing it as:
That gives the branch-cut between the singular points. Now, and . When I substitute that into the radical, I get:
Draw a picture of this and take a contour say r=3 and go completely around the origin and study how the values of and must change in order for the argument to vary smoothly around the circle: Since you want the branch that is real for real z, then we start with at the point z=3. Now go around to the point . The argument in the second quad is then but into the third quadrant, it would become , so at that point, we add the to bring it back to zero to assure continuity. Keep going around back to the point . In the fourth quad, the argument is but once it move into the first quad, it jumps back to so we then remove the to bring the argument back to zero at .
So the branch would be:
I'm describing the red sheet of the (total) real surface of this multifunction I plotted in the first plot below. Note how this sheet is contiguous and single-valued outside the branch cut.
Contrast this analysis with the function . There is no way to vary k so that the argument goes smoothly back to zero at the point in a trip around the origin. In this case, this function does NOT have analytic sheets outside the branch cut but rather consists of a single sheet wrapped around the origin 3 times as seen by the second plot. Note if you follow a circular path around the second plot, it does not return to the same point after a trip.