The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R.
H\"aring-Oldenburg, are a generalisation of the BMW algebras associated with
the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and
type B knot theory.

In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over
ground rings with parameters satisfying so-called "admissibility conditions".
These conditions are necessary in order for these results to hold and
originally arise from the representation theory of B_2^k, which is analysed by
the authors in a previous paper. Furthermore, we obtain a geometric realisation
of B_n^k as a cyclotomic version of the Kauffman tangle algebra, in terms of
affine n-tangles in the solid torus, and produce explicit bases that may be
described both algebraically and diagrammatically.

The admissibility conditions are the most general offered in the literature
for which these results hold; they are necessary and sufficient for all results
for general n.