  A Chern band is characterized by a Wannier obstruction indicating the absence
of a basis of complete, orthogonal, and exponentially-localized states. Here,
we study the properties of real space bases of a Chern band obtained by
relaxing either exponential localization or orthogonality and completeness.
This yields two distinct real space representations of a band with Chern number
$C$: (i) a basis of complete orthogonal Wannier states which decay as power-law
and (ii) a basis of exponentially-localized overcomplete non-orthogonal
coherent states. For (i), we show that the power-law tail only depends on the
Chern number and provide an explicit gauge choice leading to the universal
asymptotic $w({\boldsymbol r}) \approx \frac{C e^{-i C \varphi_{\boldsymbol
r}}}{2\pi |{\boldsymbol r}|^2}$ up to a normalized Bloch-periodic spinor. For
(ii), we prove a rigorous lower bound on the spatial spread that can always be
saturated for ideal bands. We provide an explicit construction of the maximally
localized coherent state by mapping the problem to a dual Landau level problem
where the Berry curvature and trace of the quantum metric take the roles of an
effective magnetic field and scalar potential, respectively. Our coherent state
result rigorously bounds the spatial spread of any localized state constructed
as a linear superposition of wavefunctions within the Chern band. Remarkably,
we find that such bound does not generically scale with the Chern number and
provide an explicit example of an exponentially localized state in a Chern $C$
band whose size does not increase with $|C|$. Our results show that band
topology can be encoded in a real space description and set the stage for a
systematic study of interaction effects in topological bands in real space.
