  The present article is concerned with a group of problems raised by J.
Noguchi and M. Jarnicki/P. Plug, namely whether the envelopes of holomorphy of
truncated tube domains are always schlicht, i.e. subdomains of $\mathbb{C^n}$,
and how to characterize schlichtness if this is not the case. By way of a
counter-example homeomorphic to the 4-ball, we answer the first question in the
negative. Moreover, it is possible that the envelopes have arbitrarily many
sheets. The article is concluded by sufficient conditions for schlichtness in
complex dimension two.
