  A non-unital generalization of weak bialgebra is proposed with a
multiplier-valued comultiplication. Certain canonical subalgebras of the
multiplier algebra (named the `base algebras') are shown to carry coseparable
co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier
bialgebra are shown to constitute a monoidal category via the (co)module tensor
product over the base algebra. The relation to Van Daele and Wang's (regular
and arbitrary) weak multiplier Hopf algebra is discussed.
