Examples of symmetric monoidal bicategories

Dear all,

I'm looking for examples of symmetric monoidal bicategories (where the
structure is genuinely weak, i.e. the various isomorphisms are not
identities) and I would appreciate some help. -- Actually, strict
2-categories with (genuinely) weak monoidal structure would be even more
interesting, but I found it almost impossible to find anything on that.

As I am using these categories as models, I need some structure that is
"concrete enough" to do calculations with (while being as simple as
possible).

Right now I'm considering the bicategory of rings (or monoids or
fields), bimodules over them, and bimodule homomorphisms, where the
monoidal structure is defined by the tensor product etc. (Pointers to
detailed accounts of this category would be very much appreciated, too.
I've only found fairly sketchy mentions in the literature.)

I would be grateful about any other examples of this kind!

Thanks,
Roman

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