For this question my short-hand is LP = linear program, BFS = basic feasible solution, SEF = standard equality form.
Since the Simplex algorithm iterates from extreme point to extreme point (corresponding to the fact that Simplex iterates from BFS to BFS when the LP is in SEF), how does the Simplex algorithm geometrically work when the feasible region is a polyhedron that cannot be realized in SEF (e.g. a halfspace)? Suppose we have an LP for which the feasible region has no extreme points. Then we can write an 'equivalent' LP which is in SEF and run the Simplex algorithm on it instead. But there are extreme points for this new polyhedron, whereas there are none for the original, by assumption. I originally thought that the extreme points of one LP bijectively corresponded to extreme points of the other, but this is evidently not the case.
So when exactly do the extreme points of the SEF version of an LP correspond bijectively to extreme points of the original? And further, when such a bijection does not hold, how are we supposed to geometrically interpret what the Simplex algorithm is doing in terms of the original LP?