  We derive, in the framework of soft-collinear effective field theory (SCET),
a Lagrangian describing the $t$-channel exchange of Glauber quarks in the Regge
limit. The Glauber quarks are not dynamical, but are incorporated through
non-local fermionic potential operators. These operators are power suppressed
in $|t|/s$ relative to those describing Glauber gluon exchange, but give the
first non-vanishing contributions in the Regge limit to processes such as
$q\bar q \to gg$ and $q\bar q \to \gamma \gamma$. They therefore represent an
interesting subset of power corrections to study. The structure of the
operators, which describe certain soft and collinear emissions to all orders
through Wilson lines, is derived from the symmetries of the effective theory
combined with constraints from power and mass dimension counting, as well as
through explicit matching calculations. Lightcone singularities in the
fermionic potentials are regulated using a rapidity regulator, whose
corresponding renormalization group evolution gives rise to the Reggeization of
the quark at the amplitude level and the BFKL equation at the cross section
level. We verify this at one-loop, deriving the Regge trajectory of the quark
in the $3$ color channel, as well as the leading logarithmic BFKL equation.
Results in the $\bar 6$ and $15$ color channels are obtained by the
simultaneous exchange of a Glauber quark and a Glauber gluon. SCET with quark
and gluon Glauber operators therefore provides a framework to systematically
study the structure of QCD amplitudes in the Regge limit, and derive
constraints on higher order amplitudes.
