= Solution
Let $C(z)=(2\pi i)^{-1}\int_L\phi(\tau)/(\tau-z)\,d\tau$, with interior and exterior boundary values $C_+,C_-$. The <Sokhotski–Plemelj formula> gives $\phi=C_+-C_-$ and $C_++C_-=(\pi i)^{-1}\operatorname{PV}\int_L\phi(\tau)/(\tau-t)\,d\tau$. Write $K=(\pi i)^{-1}\int_L(\tau+1/\tau)\phi(\tau)\,d\tau$ for the nonlocal scalar moment. Substitution into the <integral> equation yields
$$
2tC_+-2(t^2-1)C_-=t-1+K.
$$
Thus the associated scalar <Riemann-Hilbert problem> is
$$
\boxed{C_+(t)=\frac{t^2-1}{t}C_-(t)+\frac{t-1+K}{2t},\qquad C_-(z)=O(z^{-1})\text{ at infinity}.}
$$
The functions are analytic in their respective interior/exterior domains, and $K$ must ultimately equal the moment of the recovered jump; it is not an independently specified free constant.
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