  The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\Psi$ is
the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its
associated Galois representation. In this paper, when $\Psi$ is ordinary at $p$
we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic
measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the
Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The
construction uses techniques analogous to those used by Lei, Zerbes and the
first author in order to construct an Euler system attached to the Asai
representation of a quadratic Hilbert modular form.
