= Solution
Choose a unit vector $e$ and distinct positive numbers $a,b$, and set
$$
C_1=a(e\otimes e),
\qquad
C_2=b(e\otimes e).
$$
These are distinct rank-one covariance operators. Their positive square roots are $\sqrt a(e\otimes e)$ and $\sqrt b(e\otimes e)$, so
$$
d_R(C_1,C_2)=|\sqrt a-\sqrt b|.
$$
Using these square roots as the factors in the Procrustes definition, the identity alignment attains the same value. The singular-value formula gives the matching lower bound,
$$
d_P(C_1,C_2)^2=a+b-2\sqrt{ab}
=(\sqrt a-\sqrt b)^2.
$$
Hence $d_P=d_R$ although $C_1\ne C_2$.
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