The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces

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.