  In this work we describe horofunction compactifications of metric spaces and
finite dimensional real vector spaces through asymmetric metrics and asymmetric
polyhedral norms by means of nonstandard methods, that is, ultrapowers of the
spaces at hand. The polyhedral compactifications of the vector spaces carry the
structure of stratified spaces with the strata indexed by dual faces of the
polyhedral unit ball. Explicit neighborhood bases and descriptions of the
horofunctions are provided.
