  Based on the novel notion of `weakly counital fusion morphism', regular weak
multiplier bimonoids in braided monoidal categories are introduced. They
generalize weak multiplier bialgebras over fields and multiplier bimonoids in
braided monoidal categories. Under some assumptions the so-called base object
of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid
structure; hence to possess a monoidal category of bicomodules. In this case,
appropriately defined modules over a regular weak multiplier bimonoid are
proven to constitute a monoidal category with a strict monoidal forgetful type
functor to the category of bicomodules over the base object.
  Braided monoidal categories considered include various categories of modules
or graded modules, the category of complete bornological spaces, and the
category of complex Hilbert spaces and continuous linear transformations.
