How to choose a phase for the deconvolution of an autocorrelation?
For a real-valued signal $C$, the autocorrelation function $M$ is a real-valued even
function and the power spectral density $m$ (Fourier transform of $M$) is a real-valued
nonnegative even function.  Now, $m(k) = c(k)c^*(k)$ where $c$ is the Fourier transform
of $C$, as you correctly assert, but given only $m$ and no other
information about $c$ (or $C$),
it is not possible to determine $c$ at all.  Note that $m(k)$ also equals $[c(k)e^{j\theta_k}][c(k)e^{j\theta_k}]^* = c(k)c^*(k) e^{j\theta_k}e^{-j\theta_k}$ for
arbitrary choice of $\theta_k$, and the inverse Fourier transform of $c(k)e^{j\theta_k}$
is also a real-valued signal if we are careful to preserve conjugate symmetry so
that we choose $\theta_{-k}$ as $-\theta_k$. In other words, there are infinitely many
quite different signals that share the same autocorrelation function, and it is not
the case that only $C$ and its time-delayed versions have autocorrelation function $M$.
As an example, all PN signals, that is, pulse trains generated from a maximal-length
linear feedback shift register (LFSR), have an inverted thumbtack autocorrelation,
and the distinct PN signals (from different LFSRs)
are even two-level signals, a property not shared
by all of the infinitely many real signals that have inverted
thumbtack autocorrelation.