  In this paper, we associate Galois representations to globally generic
cuspidal automorphic representations on GSp(4), over a totally real field F,
which are Steinberg at some finite place. This association is compatible with
the local Langlands correspondence for GSp(4) studied recently in a preprint of
Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at
p to the dimensions of the parahoric fixed spaces at p. The Galois
representations are constructed by first passing to GL(4) over a CM extension,
then applying the book of Harris-Taylor plus a refinement due to
Taylor-Yoshida, and finally descending to F by a delicate patching argument.
This is a variation of the techniques used by Blasius and Rogawski in order to
attach motives to Hilbert modular forms.
