Comments to the question (v1):
I) Reconstruction of phases from modulus$^1$ $|f(x)|$ of a signal $f(x)$ and modulus $|\tilde{f}(k)|$ of its Fourier transformed (FT) signal
$$\tag{1} \tilde{f}(k) ~:=~ \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \!dx~ e^{-ikx} f(x)$$
is an interesting and likely a well-studied engineering problem, either for continuous or discrete Fourier transformation.
II) Example: A Gaussian signal
$$\tag{2} f(x)~=~Ae^{-\frac{a}{2}x^2+bx}, \qquad A,a,b\in \mathbb{C}, \qquad
{\rm Re}(a)~>~0, $$
with Fourier transform
$$\tag{3} \tilde{f}(k)~=~\frac{A}{\sqrt{a}}\exp\left[-\frac{(k+ib)^2}{2a}\right]
~=~\frac{A}{\sqrt{a}}\exp\left[-\frac{\bar{a}}{2|a|^2}(k+ib)^2\right]
~=~\frac{A}{\sqrt{a}}\exp\left[-\frac{\bar{a}}{2|a|^2}(k^2+2ibk-b^2)\right];$$
with modulus
$$\tag{4} |f(x)| ~=~ |A| \exp\left[-\frac{{\rm Re}(a)}{2}x^2+{\rm Re}(b)x\right], $$
and
$$\tag{5} |\tilde{f}(k)| ~=~ \frac{|A|}{\sqrt{|a|}}
\exp\left[-\frac{{\rm Re}(a)k^2-2{\rm Im}(\bar{a}b)k-{\rm Re}(\bar{a}b^2)}{2|a|^2}\right], $$
respectively. It is interesting that if one additionally knows that the signal
is of Gaussian form (2), then it is possible from (4) and (5) to reconstruct the constant $a$ up to possibly a sign ambiguity of ${\rm Im}(a)$; the constant $A$ up to a phase; and the constant $b$ is unique for given choice of $a$.
III) The above Gaussian example induces hope that modulus of a signal and modulus of its FT are sufficiently complementary information such that reconstruction is possible up to possibly a finite number of self-consistent solutions, and modulo an overall global phase.
IV) We speculate that it may in practice be possible to reconstruct a self-consistent signal from its modulus and modulus of its FT via an iterative fixed-point algorithm$^2$: First Fourier transform the bare modulus of the signal; next multiply the phases of the result with the initially given FT modulus; then inverse FT; next multiply the phases of the result with the initially given modulus; then FT; and so forth, until a self-consistent fixed-point configuration is reached on each side of the FT.
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$^1$ In signal analysis the modulus is also called amplitude, magnitude, or absolute value.
$^2$ Update: It turns out that this algorithm exists and is known as Gerchberg-Saxton algorithm (hat tip: WetSavannaAnimal aka Rod Vance).