Truncated tube domains with multi-sheeted envelope

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.