= Solution
Apply the reciprocal identity with flow1 the actual body-force-driven flow and flow2 the <rotlet> auxiliary field. Its <body force> is zero. The large-sphere boundary terms vanish under the usual decay and convergence assumptions for this unbounded body-force problem.
On the <sphere>, the actual rigid <velocity> is $\mathbf U+\boldsymbol\Omega\times\mathbf x$. The auxiliary <force> is zero and its <torque> with the outward fluid normal is $\mathbf G$. The actual <torque> with that normal is the negative of the <hydrodynamic torque> on the <sphere>, hence zero for a <torque-free> <sphere>. Consequently the surface side of the reciprocal identity is just $\boldsymbol\Omega\cdot\mathbf G$. The volume side is
$$
\int_{r>a}\mathbf u_G\cdot\mathbf f\,dV=\frac{\mathbf G}{8\pi\mu}\cdot\int_{r>a}\frac{\mathbf x\times\mathbf f(\mathbf x)}{r^3}dV.
$$
Since the auxiliary <couple> is arbitrary,
$$
\boxed{\boldsymbol\Omega=\frac1{8\pi\mu}\int_{r>a}\frac{\mathbf x\times\mathbf f(\mathbf x)}{r^3}dV.}
$$
An unknown translation contributes nothing because the auxiliary <rotlet> has zero net <force>. This is <body-force-driven rotation of a torque-free sphere>.
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