  The upper critical field of multi-band superconductors can be an essential
quantity to unravel the nature of superconducting pairing and its interplay
with the electronic structure. Here we experimentally map out the complete
upper critical field phase diagram of FeSe for different magnetic field
orientations at temperatures down to 0.3 K using both resistivity and torque
measurements. The temperature dependence of the upper critical field reflects
that of a multi-band superconductor and requires a two-band description in the
clean limit with band coupling parameters favouring interband over intraband
interactions. Despite the relatively small Maki parameter in FeSe of $\alpha
\sim 1.6$, the multi-band description of the upper critical field is consistent
with the stabilization of a FFLO state below $T/T_{\rm c}\sim 0.3$. We find
that the anomalous behaviour of the upper critical field is linked to a
departure from the single-band picture, and FeSe provides a clear example where
multi-band effects and the strong anisotropy of the superconducting gap need to
be taken into account.
