  A numerical description of an algebraic subvariety of projective space is
given by a general linear section, called a witness set. For a subvariety of a
product of projective spaces (a multiprojective variety), the corresponding
numerical description is given by a witness collection, whose structure is more
involved. We build on recent work to develop a toolkit for the numerical
manipulation of multiprojective varieties that operates on witness collections,
and use this toolkit in an algorithm for numerical irreducible decomposition of
multiprojective varieties. The toolkit and decomposition algorithm are
illustrated throughout in a series of examples.
