There is a known power spectrum $$C_l=\frac1{2l+1}\sum_{m=-l}^l |a_{lm}|^2$$ for some $l=0,1,\dots,l_{max}$ where $$ a_{lm}=\int_{-\pi/2}^{\pi/2} \mathrm \int_0^{2\pi}f(\theta,\phi)Y^*_{lm}(\theta,\phi)\,d\phi\,d\theta$$ are got using complex spherical harmonics for an unknown real function $f$. Is there a way to "restore" such a sample function $f$?
The problem is that besides $f$ is represented as $$f(\theta,\phi)=\sum_{l=0}^{l_{max}}\sum_{m=-l}^l a_{lm}Y_{lm}(\theta,\phi),$$ taking arbitrary complex $a_{lm}$ with known sums of absolute values for each $l$ could produce a complex $f$ while a real function is required.
Is there a property of $a_{lm}$ coefficients for real $f$ or some condition on $a_{lm}$ that guarantee $f$ to be real?
UPD. Using real spherical harmonics could be a solution, but they probably produce a different power spectrum. So, the task could be solved if there is any relation between the real and complex power spectra.