  Taking the signature of the closure of a braid defines a map from the braid
group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism
defect of this map in terms of the Meyer cocycle and the Burau representation.
In the present paper, we simultaneously extend this result in two directions,
considering the multivariable signature of the closure of a colored tangle. The
corresponding defect is expressed in terms of the Maslov index and of the
Lagrangian functor defined by Turaev and the first-named author.
