The Schroedinger equation can describe a charged particle moving
in some given electric field; the equation is linear in the
wavefunction. A charged particle, being charged, also produces its
own electric field - this 'self-field' is nothing magical, it is
the same old field that is responsible for scattering other charged
particles. But in quantum mechanics we do not ordinarily include a
charged particle's generation of and reaction to its own field (it
is included in some sense in quantum field theory). If we consider
the whole electric field (i.e. do not ignore the self-field) the
Schroedinger equation + Maxwell's equations together are nonlinear.

This matters because the measurement problem follows fairly directly
from the linearity of the Schroedinger equation, but if the full
set of equations is nonlinear then the straightforward argument
from linearity to the measurement problem doesn't get off the ground.

Another consequence is that the spreading of a free wavepacket is
not obviously physically relevant for electrons; an electron generates
a strong electric field (and magnetic field from its motion and
from the current associated with its spin), and its interaction
with these fields will certainly affect its dynamics.

It isn't super-easy to look at this with quantum field theory since
the everyday versions of QFT are concerned primarily with scattering;
they give us amplitudes for various scattering processes between
free particle states but the state of the particles as they interact,
or even of the particles just surrounded by their own electric
fields, isn't readily available (in the very simplest versions of
QFT at least - I'm certainly no expert!).

I mean to do some numerical simulations of the Schroedinger equation
coupled to Maxwell's equations (or maybe a toy version) just to see
how this looks.