  Manifolds with fibered corners arise as resolutions of stratified spaces, in
many body compactifications of vector spaces, moduli spaces, and other
settings. We define a category of fibered corners manifolds which has products
and transverse fiber products, generalizing both the resolutions of products of
stratified spaces and many body products, which are special cases. The product
in the fibered corners category is a resolution of the cartesian product by
blow-up which we call the 'ordered product'. This ordered product is a natural
product for wedge (aka incomplete edge) metrics and quasi-fibered boundary
metrics, a class which includes QAC and QALE metrics.
