  Let $(\Omega,g)$ be a smooth compact two-dimensional Riemannian manifold with
boundary, $\Lambda_g: f\mapsto \partial_\nu u|_{\partial\Omega}$ its DN map,
where $u$ obeys $\Delta_g u=0$ in $\Omega$ and $u|_{\partial \Omega}=f$. The
Electric Impedance Tomography problem is to determine $\Omega$ from
$\Lambda_g$.
  A criterion is proposed that enables one to detect (via $\Lambda_g$) whether
$\Omega$ is orientable or not.
  The algebraic version of the BC-method is applied to solve the EIT problem
for the Moebius band. The main instrument is the algebra of holomorphic
functions on the double covering ${\mathbb M}$ of $M$, which is determined by
$\Lambda_g$ up to an isometric isomorphism. Its Gelfand spectrum (the set of
characters) plays the role of the material for constructing a relevant copy
$(M',g')$ of $(M,g)$. This copy is conformally equivalent to the original,
provides $\partial M'=\partial M,\,\,\Lambda_{g'}=\Lambda_g$, and thus solves
the problem.
