Habemus plus vis computatoris quam Deus

I refer you back to this post on the "hall of mirrors" universe.

Consider, now, this:

Imagine that you and a friend build two identical relativistic starships, and set off across the Universe on parallel courses, a short distance apart, at 0.999999C.  Assume that you have some method, be it optical, gravitational or whatever, of visually observing the other's ship, flying a few miles away from you.  In only a few years, ship time, you reach an "edge", or even more interestingly a vertex, of the universe, and you "wrap around" and transition to the opposing "facet" of the four-dimensional space.  Suppose, further, that your parallel courses happen to hit the edge - or vertex - so perfectly that you transition through one "facet", while your friend transitions through a different, adjacent "facet" touching yours only at an edge or vertex.

What happens?  Is your friend's ship still flying next to you?  Or does it, from your perspective, vanish (and you from your friend's), the two ships completing their transition billions of light years apart?  Is it possible to map the faces in such a way that, for all possible edges and vertices, points that are adjacent on one side of a "wrap transition", but on separate facets, are also adjacent on the far side of the transition, even though both faces have been "rotated" through 36 degrees?

Note:  This is almost certainly a trick question ... I think.  My intuition on this is that, from the perspective of an observer in three-dimensional space, you would always be somewhere within the body of the Poincaré dodecahedral space - or whatever shape the manifold turns out to have - and would not be able to actually approach the edges or vertices of the four-dimensional space.  (Remember, too, that we're not talking about a dodecahedral three-dimensional space, we're talking about an apparently-unbounded three-dimensional space mapped onto a four-dimensional dodecahedral space.)  But I haven't done the math - I don't even begin to know how to do the math - and so I could be wrong.  You never know, it could turn out that some related effect is responsible for cosmic megastructures such as the Great Wall or the Great Void.