= Solution
Normalization of a <wavefunction> on the unit sphere uses solid angle $d\Omega=\sin\theta\,d\theta\,d\phi$, not the flat measure $d\theta\,d\phi$. Since $|e^{-i\phi}|=1$,
$$
\int|\psi|^2d\Omega=\int_0^{2\pi}d\phi\int_0^\pi\sin^3\theta\,d\theta
=2\pi\int_{-1}^{1}(1-u^2)\,du=\frac{8\pi}{3},
$$
where $u=\cos\theta$. Therefore
$$
\boxed{\Psi(\theta,\phi)=\sqrt{\frac3{8\pi}}\sin\theta\,e^{-i\phi}.}
$$
With the usual phase convention this is the <spherical harmonic> $Y_{1,-1}$. An arbitrary global unit-modulus factor would not change normalization or any observable.
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