I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of C*-algebras.

In the algebraic version, we are interested in the monoid structure of the Morita equivalence classes of $R$-algebras (where R is a commutative ring), given by the tensor product over $R$. In particular, the invertible elements of this monoid are given by the Azumaya algebras over $R$, and they form the Brauer group of $R$. Do similar phenomena occur for C*-algebras?

For von Neumann algebras, I asked a question on Math.SE, and someone commented that the theory is basically empty: since factors of type I are Morita-trivial, and a tensor product of a factor of type II or III with another factor is again of type II or III, in the end the only way to be Morita-invertible is to be Morita-trivial.

Do C*-algebras offer more theory? I understand that in that context one has to be a little more careful: the Morita-equivalence I care about is the strong Morita equivalence (as defined by imprimitivity bimodules). Also, talking about tensor products can be awkward, so maybe I should restrict to nuclear C*-algebras (but if there are results of Morita-invertibility for well-chosen tensor products on non-nuclear algebras, I am also interested in hearing about it).

Clearly, one has to restrict to unital and central algebras, so in the end my question is the following:

If $A$ is a central unital C*-algebra (maybe nuclear), and there exists $B$ such that $A\otimes B$ is strongly Morita-equivalent to $\mathbb{C}$ (for some tensor product if $A$ is not nuclear), does it follow that $A$ itself is strongly Morita-equivalent to $\mathbb{C}$?

I am also interested in similar results for real C*-algebras (maybe even more so).