  We propose a preconditioner for the Helmholtz exterior problems on
multi-screens. For this, we combine quotient-space BEM and operator
preconditioning. For a class of multi-screens (which we dub \emph{type A}
multi-screens), we show that this approach leads to block diagonal Calder\'on
preconditioners and results in a spectral condition number that grows only
logarithmically with $h$, just as in the case of simple screens. Since the
resulting scheme contains many more DoFs than strictly required, we also
present strategies to remove almost all redundancy without significant loss of
effectiveness of the preconditioner. We verify these findings by providing
representative numerical results.
  Further numerical experiments suggest that these results can be extended
beyond type A multi-screens and that the numerical method introduced here can
be applied to essentially all multi-screens encountered by the practitioner,
leading to a significantly reduced simulation cost.
