Not an answer but too large for a comment.

In my paper with van de Lune *On the exact location of the non-trivial zeros f Riemann's zeta function, it is proved*.

**Theorem.** If $f\colon\mathbf{R}\to\mathbf{C}$ is real analytic, then there are two real analytic functions
$U\colon\mathbf{R}\to\mathbf{R}$ and $\varphi\colon\mathbf{R}\to\mathbf{R}$ such that $f(t)=U(t)e^{i\varphi(t)}$. Given two such representations, $f=U_1e^{i\varphi_1}$ and $f=U_2e^{i\varphi_2}$ we have either $U_1=U_2$ and $\varphi_1-\varphi_2=2k\pi$ or $U_1=-U_2$ and $\varphi_1-\varphi_2=(2k+1)\pi i$ for some integer $k$.

Your question is slightly different. If a real analytic function can be represented as $f(x)=A(x)\sin(\phi(x))$ with $A$ and $\phi$ real analytic, then $f(x)=\Im (A(x)e^{i\phi(x)})$. The function $F(x) =A(x)e^{i\phi(x)}$ will be a complex real analytic function with $f(x)=\Im(F(x))$. But this $F$ is not unique. Any real and real analytic function $h(x)$ gives us
$F(x)+h(x)$ with the same property. By the above theorem we will have
$F(x)+h(x)=B(x)e^{i\phi_h(x)}$ with $B$ and $h$ real real analytic functions. Therefore we obtain
$f(x)=B(x) \sin(\phi_h(x))$. All representation to your functions are of this type, starting from the one given in the comment by Carlo Beenakker.

The problem whether we may get $\phi_h(x)$ completely monotone appear to be complicated. In simple cases as $\Gamma(s/2)=|\Gamma(s/2)|e^{i\vartheta(t)+i\frac{t}{2}\log\pi}$, we get the representation
$$\Im(\Gamma(\tfrac14+i\tfrac{t}{2}))=|\Gamma(\tfrac14+i\tfrac{t}{2})|\sin(\vartheta(t)+\tfrac{t}{2}\log\pi),\qquad s=\tfrac12+it$$
the phase is not monotonous for $t>0$. Certainly $\vartheta(t)+\frac{t}{2}\log\pi$ appear to be almost completely monotonous. I will be very surprised if in this case there is a completely monotonous phase.