  The tenfold way provides a strong organizing principle for invertible
topological phases of matter. Mathematically, it is intimately connected with
$K$-theory via the fact that there exist exactly ten Morita classes of simple
real superalgebras. This connection is physically unsurprising, since weakly
interacting topological phases are classified by $K$-theory. We argue that when
strong interactions are present, care has to be taken when formulating the
exact ten symmetry groups present in the tenfold way table. We study this
phenomenon in the example of class D by providing two possible mathematical
interpretations of a class D symmetry. These two interpretations of class D
result in Morita-equivalent but different symmetry groups. As $K$-theory cannot
distinguish Morita-equivalent protecting symmetry groups, the two approaches
lead to the same classification of topological phases on the weakly interacting
side. However, we show that these two different symmetry groups yield different
interacting classifications in spacetime dimension 2+1. We use the approach to
interacting topological phases using bordism groups, reducing the relevant
classification problem to a spectral sequence computation.
