The way to answer this question is to write the quantities in covariant form. I use Heaviside-Lorentz units as usual, which makes things most convenient. Electromagnetically the matter is characterized by charge and current densities as well as electric and magnetic polarization densities. The charge and current densities are combined to a four-vector,
$$j^{\mu}=\begin{pmatrix} c \rho \\ \vec{j} \end{pmatrix}.$$
The polarizations are a bit more cumbersome to explain. First one must remember that the Maxwell equations in matter are written in a form which splits the fields in parts created by the free sources and polarization fields resulting from intrinsic charge, current, and magnetization distributions. The total electromagnetic field ##(\vec{E},\vec{B})## and the ones created by the free charges and currents ##(\vec{D},\vec{H})##. These (and only these combinations) build antisymmetric tensors
$$(F_{\mu \nu}) \equiv (\vec{E},\vec{B}) = \begin{pmatrix} 0 & -E_x & -E_y & - E_x \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{pmatrix}.$$
Analogously one has
$$(D_{\mu \nu})=(\vec{D},\vec{H}).$$
Now for a historical reason and a misunderstanding about which fields combine naturally together the definition of the electric and magnetic polarization are different in sign, i.e., one defines
$$\vec{E}=\vec{D}+\vec{P}, \quad \vec{B}=\vec{H} - \vec{M},$$
and that's why the polarization tensor combines to
$$P_{\mu \nu} =(\vec{P},-\vec{M}).$$
Despite this confustion with the signs, now it's easy to understand what's going on concerning how different observers see the electromagnetic field of a permanent magnet.
In its rest frame it's uncharged, and there are no currents, i.e., in the rest frame you have
$$\rho=0, \quad \vec{j}=0 \; \Rightarrow \; j^{\mu}=0.$$
Under Lorentz boosts this transforms like a Minkowski vector, and thus
$$j^{\prime \mu}={\Lambda^{\mu}}_{\nu} j^{\nu}=0.$$
Also for the observer relative to which the magnet is moving with constant velocity, there are neither charge nor current densities.
For the observer at rest, we can also assume that ##\vec{P}=0## and ##\vec{M} \neq 0## (i.e., it's electrically unpolarized, but it has a magnetization of course). Now the Polarization tensor transforms as any 2nd-rank tensor,
$$P^{\prime \mu \nu}={\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma} P^{\rho \sigma},$$
and this gives you entries also for the electric polarization, i.e., in the reference frame, where the magnet is moving, there's both an electric polarization and magnetization, and that's why in this frame there's also a electric field, while in the frame where the magnet is at rest there's only a magnetic field. In other words, a body which has only a magnetization when viewed in its rest frame becomes also electrically polarized.
