  For a stochastic process $\{X_t\}_{t \in T}$ with identical one-dimensional
margins and upper endpoint $\tau_{\text{up}}$ its tail correlation function
(TCF) is defined through $\chi^{(X)}(s,t) = \lim_{\tau \to \tau_{\text{up}}}
P(X_s > \tau \,\mid\, X_t > \tau )$. It is a popular bivariate summary measure
that has been frequently used in the literature in order to assess tail
dependence. In this article, we study its realization problem. We show that the
set of all TCFs on $T \times T$ coincides with the set of TCFs stemming from a
subclass of max-stable processes and can be completely characterized by a
system of affine inequalities. Basic closure properties of the set of TCFs and
regularity implications of the continuity of $\chi$ are derived. If $T$ is
finite, the set of TCFs on $T \times T$ forms a convex polytope of $\lvert T
\rvert \times \lvert T \rvert$ matrices. Several general results reveal its
complex geometric structure. Up to $\lvert T \rvert = 6$ a reduced system of
necessary and sufficient conditions for being a TCF is determined. None of
these conditions will become obsolete as $\lvert T \rvert\geq 3$ grows.
