  We introduce tropical spectrahedra, defined as the images by the
nonarchimedean valuation of spectrahedra over the field of real Puiseux series.
We provide an explicit polyhedral characterization of generic tropical
spectrahedra, involving principal tropical minors of size at most 2. One of the
key ingredients is Denef-Pas quantifier elimination result over valued fields.
We obtain from this that the nonarchimedean valuation maps semialgebraic sets
to semilinear sets that are closed. We also prove that, under a regularity
assumption, the image by the valuation of a basic semialgebraic set is obtained
by tropicalizing the inequalities which define it.
