Is there a formula for the surgery obstruction of a fiber product of maps assuming the fiber product is also a homotopy equivalence?

In more detail, suppose that $X to Y$ and $Z to Y$ are homotopy equivalences of (topological, say, compact oriented) manifolds. There is a long history of product formulae for surgery obstructions which includes a surgery obstruction for the product map $X times Z to Y times Y$ to be homotopic to a homeomorphism. See for example Section 8 in Ranicki, Algebraic Theory of Surgery II. Applications to Topology.

Suppose I know that the fiber product $X times_Y Z to Y$ is also a homotopy equivalence. (Often it won’t be, but in my case I got lucky.) Is there a formula for the surgery obstruction of this map in terms of those of $X to Y$ and $Z to Y$? I am actually just interested in the case $X times_Y X to Y$, perturbed to make the fiber product transverse and would like to know what are the possibilities for the surgery obstruction of this map. (Could it be anything?)