  The tail-dependence compatibility problem is introduced. It raises the
question whether a given $d\times d$-matrix of entries in the unit interval is
the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random
vector. The problem is studied together with Bernoulli-compatible matrices,
that is, matrices which are expectations of outer products of random vectors
with Bernoulli margins. We show that a square matrix with diagonal entries
being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible
matrix multiplied by a constant. We introduce new copula models to construct
tail-dependence matrices, including commonly used matrices in statistics.
