Monoidal structures on von Neumann algebras

The category of von Neumann algebras W* admits a variety of monoidal structures of three distinct flavors.

(1) W* is complete and therefore you have a monoidal structure given by the categorical product.

(2a) W* is cocomplete and therefore you also have a monoidal structure given by the categorical coproduct.

(2b) I suspect that there is also a “spatial coproduct”, just as there is a categorical
tensor product and a spatial tensor product (see below).
The spatial coproduct should correspond to a certain central projection in the categorical coproduct.
Perhaps the spatial coproduct is some sort of coordinate-free version
of the free product mentioned in Dmitri Nikshych's answer.

(3a) For any two von Neumann algebras M and N
consider the functor F from W* to Set that sends a von Neumann algebra L
to the set of all pairs of morphisms M→L and N→L with commuting images.
The functor F preserves limits and satisfies the solution set condition, therefore it is representable.
The representing object is the categorical tensor product of M and N.

(3b) There is also the classical spatial tensor product.
I don't know any good universal property that characterizes it except
that there is a canonical map from (3a) to (3b) and its kernel corresponds
to some central projection in (3a). Perhaps there is a nice description of this central projection.

Since your monoidal structure is of the third flavor and you don't want a monoidal
structure of the first flavor, I suggest that you try a monoidal structures of the second flavor.
I suspect that the spatial coproduct of two factors is actually a factor.
You are lucky to work with factors, because in the commutative case 2=3, in particular 2a=3a and 2b=3b.