In the case of splitting traffic between two hosts depending on some factor, what if you want outgoing requests from those services to appear to come from the same place (so the two machine appear to be one interface for outgoing connections as well as incoming ones)?

You could use some sort of proxy services the the machines make the requests through (and the passes incoming requests back - you can avoid NAT in both directions that way) of course, but NAT seems (to me) to be a lighter solution for that use case.