Suppose that a matrix
$$A ~=~ x_1 B + x_2 C$$
is a linear combination of two self-adjoint matrices $B$ and $C$.
I'm interested in when $A$ represents a physical quantity.
When the linear combination is a complex combination, then $B$ and $C$ have to be commutable for $A$ to represent any physical quantity, cf. this Phys.SE post.
Now suppose that $x_1$ and $x_2$ are real. What happens in this case? If $B$ and $C$ are noncommutable, does $A$ still represent physical quantity?