  In this work, Stroh's formalism is endowed with causal properties on the
basis of an analysis of the radiation condition in the Green tensor of the
elastodynamic wave equation. The modified formalism is applied to dislocations
moving uniformly in an anisotropic medium. In practice, accounting for
causality amounts to a simple analytic continuation procedure whereby to the
dislocation velocity is added an infinitesimal positive imaginary part. This
device allows for a straightforward computation of velocity-dependent field
expressions that are valid whatever the dislocation velocity ---including
supersonic regimes--- without needing to consider subsonic and supersonic cases
separately. As an illustration, the distortion field of a Somigliana
dislocation of the Peierls-Nabarro-Eshelby-type with finite-width core is
computed analytically, starting from the Green tensor of elastodynamics. To
obtain the result in the form of a single compact expression, use of the
modified Stroh formalism requires splitting the Green function into its
reactive and radiative parts. In supersonic regimes, the solution obtained
displays Mach cones, which are supported by Dirac measures in the Volterra
limit. From these results, an explanation of Payton's `backward' Mach cones
[R.G. Payton, Z. Angew. Math. Phys. 46, 282--288 (1995)] is given in terms of
slowness surfaces, and a simple criterion for their existence is derived. The
findings are illustrated by full-field calculations from analytical formulas
for a dislocation of finite width in iron, and by Huygens-type geometric
constructions of Mach cones from ray surfaces.
