  We define a family of generalizations of $\operatorname{SL}_2$-tilings to
higher dimensions called $\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings.
We show that, in each dimension 3 or greater,
$\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings exist only for certain
choices of $\boldsymbol{\epsilon}$. In the case that they exist, we show that
they are essentially unique and have a concrete description in terms of odd
Fibonacci numbers.
