How to choose a phase for the deconvolution of an autocorrelation? Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where $\chi\left(x\right)$ is some complex valued basis function.

I know that the autocorrelation is given by $M\left(\Delta x\right)=\sum_{k=-\infty}^{\infty}m\left(k\right)\chi\left(\Delta x\right)$, where $\Delta x = x_2-x_1$ and by the convolution theorem I know that $m\left(k\right)=\left(c\left(k\right)\right)^*c\left(k\right)$. This means that knowing the coefficients of the autocorrelation immediately gives me the squared magnitude of the original coefficients. This implies that any two functions $C'\left(x\right)=C\left(x-x'\right)$, that differ only by a translation, will possess the same autocorrelation, $M\left(\Delta x\right)$, and so if I am trying to obtain the $c\left(k\right)$ from the $m\left(k\right)$, then I can choose an arbitrary phase for the $c\left(k\right)$. However, I don't think I can choose the phase of $c\left(k\right)$ independently for each $k$, it seems like there needs to be some way to consistently choose the phase for all $k$ (which corresponds to specifying a translation for $C\left(x\right)$).

My question is: How do I choose a consistent phase for each of the coefficients for the original function, i.e. for all $k$?
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.