  The Goldman-Iwahori space of seminorms on a finite-dimensional vector space
over a non-Archimedean field is a non-Archimedean analogue of a symmetric
space. If, in addition, $K$ is real closed, we define a signed analogue of the
Goldman-Iwahori space consisting of signed seminorms. This new space can be
seen as the linear algebraic version of the real analytification of projective
space over $K$. We study this space with methods from real tropical geometry by
constructing natural real tropicalization maps from the signed Goldman-Iwahori
space to all real tropicalized linear spaces. We prove that this space is the
limit of all real tropicalized linear embeddings. We give a combinatorial
interpretation of this result by showing that the signed Goldman-Iwahori space
is the real tropical linear space associated to the universal realizable
oriented matroid. In the constant coefficient case for $K = \mathbb{R}$, we
describe this space explicitly and relate it to real Bergman fans.
