  We study the electromagnetic coupling of a neutrino that propagates in a
two-stream electron background medium. Specifically, we calculate the
electromagnetic vertex function for a medium that consists of a \emph{normal}
electron background plus another electron \emph{stream} background that is
moving with a velocity four-vector $v^\mu$ relative to the normal background.
The results can be used as the basis for studying the neutrino electromagnetic
properties and various processes in such a medium. As an application, we
calculate the neutrino dispersion relation in the presence of an external
magnetic field ($\vec B$), focused in the case in which $B$ is inhomogeneous,
keeping only the terms of the lowest order in $1/m^2_W$ and linear in the $B$
and its gradient. We show that the dispersion relation contains additional
anisotropic terms involving the derivatives of $\vec B$, such as the gradient
of $\hat k\cdot(\vec v\times\vec B)$, which involve the stream background
velocity, and a term of the form $\hat k\cdot(\nabla\times \vec B)$ that can be
present in the absence of the stream background, in addition to a term of the
form $\hat k\cdot\vec v$ and the well known term $\hat k\cdot\vec B$ that
arises in the constant $\vec B$ case. The derivative-dependent terms are even
under a $CP$ transformation. As a result, in contrast to the latter two just
mentioned, they depend on the sum of the particle and antiparticle densities
and therefore can be non-zero in a $CP$-symmetric medium in which the particle
and antiparticle densities are equal.
