  This paper introduces new tests of fundamental physics by means of the
analysis of disturbances on the GNSS signal propagation. We show how the GNSS
signals are sensitive to a space variation of the fine structure constant
$\alpha$ in a generic framework of effective scalar field theories beyond the
Standard Model. This effective variation may originate from the crossing of the
RF signals with dark matter clumps and/or solitonic structures. At the
macroscopic scale, the subsequent disturbances are equivalent to those which
occur during the propagation in an inhomogeneous medium. We thus propose an
interpretation of the "measure" of the vacuum permeability as a test of
fundamental physics. We show the relevance of our approach by a first
quantification of the expected signature in a simple model of a variation of
$\alpha$ according to a planar geometry. We use a test-bed model of domain
walls for that purpose and focus on the measurable time delay in the GNSS
signal carrier.
