Review of Kato–Nakayama spaces
If $X$ is an fs (finite and saturated) log scheme such that the underlying scheme $\underline X$ is of finite type over $\mathbf{C}$, one can associate to it in a functorial way a topological space $X^{\rm log}$, called the Kato–Nakayama space of $X$, together with a proper map $\tau:X^{\rm log}\to X$. The points of $X^{\rm log}$ correspond to pairs $(x, h)$ of a point $x\in\underline X$ and a map $h:\mathscr{M}_{X, x}\to \mathbf{S}^1$ such that for $f\in \mathscr{O}^\times_{X, x}$, $h(f) = f(x)/|f(x)|$.
In the special case when $X=\mathbf{A}^1$ with the usual log structure given by the open immersion $j:\mathbf{G}_m\hookrightarrow\mathbf{A}^1$, $X^{\rm log}$ is the `real blow-up' $$ X^{\rm log} = \mathbf{R}_{\geq 0}\times \mathbf{S}^1 $$ and the map $\tau$ sends $(r, \theta)$ to $r\cdot\theta\in \mathbf{C}=\mathbf{A}^1(\mathbf{C})$. The open immersion $j$ factors through $\tau$, and the fiber $\tau^{-1}(0)\cong \mathbf{S}^1$.
In general, the Kato-Nakayama space has the following nice properties:
If $f:X\to Y$ is strict (i.e., the induced $f^*\mathscr{M}_Y\to \mathscr{M}_X$ is an isomorphism), then the square $\require{AMScd}$ \begin{CD} X^{\rm log} @>>> Y^{\rm log} \\ @VVV @VVV \\ X @>>> Y \\ \end{CD} is cartesian.
If $X=\mathbf{A}_P$ for an fs monoid $P$ (by definition, $\mathbf{A}_P={\rm Spec}(P\to \mathbf{C}[P])$), then $$ X^{\rm log} = {\rm Hom}(P, \mathbf{R}_{\geq 0}\times \mathbf{S}^1 ) $$ and the map $\tau$ is induced by the multiplication $\mathbf{R}_{\geq 0}\times \mathbf{S}^1 \to \mathbf{C}$. In particular, $X^{\rm log}$ is a manifold with boundary. Since an fs log scheme is by definition one which locally admits a strict map to some $\mathbf{A}_P$, this together with 1. gives a complete local description of $X^{\rm log}$.
The fiber over $x\in \underline X$ is a torsor under ${\rm Hom}(\mathscr{M}_{X, x}, \mathbf{S}^1)$ in a natural way.
If $X$ is log regular (say, log smooth over $\mathbf{C}$ with the trivial log structure), then $X^{\rm log}$ is a manifold with boundary. If $X_{\rm tr}$ denotes the biggest open of $\underline X$ where the log structure is trivial, then the inclusion $j:X_{\rm tr}\hookrightarrow \underline X$ lifts to an open immersion $\bar j : X_{\rm tr}\hookrightarrow X^{\rm log}$ along $\tau$. Moreover, $\bar j$ is a homotopy equivalence.
(Nakayama–Ogus) If $f:X\to Y$ is log smooth and exact, then the induced $f^{\rm log}:X^{\rm log}\to Y^{\rm log}$ is a topological submersion (i.e., locally on the source looks like a projection), and its fibers are manifolds with boundary. If $\underline f$ is proper, then $f^{\rm log}$ is a locally trivial fibration.
Notation. $V=\mathbf{C}[[t]]$, $K = {\rm Frac}(V)= \mathbf{C}((t))$, $\bar K = \bigcup_n \mathbf{C}((t^{1/n}))$ the algebraic closure of $K$, $k=V/m = \mathbf{C}$, $S={\rm Spec}\, V$, $\eta = {\rm Spec}\, K$, $\bar\eta = {\rm Spec}\, \bar K$, $s={\rm Spec}\, k$. We give $S$ the natural log structure coming from the open immersion $\eta\hookrightarrow S$.
Let $C$ be the category of smooth schemes over $\eta$, and let $\bar C$ be the category of proper log smooth fs log schemes over $S$. We have two crucial functors: $$ X\mapsto X_{\rm tr} : \bar C \to C $$ $$ X\mapsto X_s^{\rm log} : \bar C \to {\rm Top}_{/\mathbf{S}^1} $$ The first one associates to a log scheme $X/S$ the locus $X_{\rm tr}$ where the log structure is trivial. This maps to $S_{\rm tr} = \eta$, so is an $\eta$-scheme, and it is smooth when $X$ is log smooth. The second functor is the Kato–Nakayama space of the special fiber $X_s$ with the induced log structure, endowed with the natural map $X_s^{\rm log} \to s^{\rm log} \cong \mathbf{S}^1$. It is a locally trivial fibration whose fibers are manifolds with boundary (by Nakayama-Ogus).
The functors have the following nice properties:
Suppose that an object $X$ of $\bar C$ arises as a base change of a proper log smooth analytic space $X$ over an open disc $\Delta$ with log structure induced by $\Delta\setminus\{0\}$. Then $Y^{\rm log}\to U^{\rm log}$ is a locally trivial fibration (by Nakayama–Ogus) and $X^{\rm log}_s\to s^{\rm log}$ is its base change to the base point $s=0$. So in this `convergent' case the construction recovers what you want.
The functor $X\mapsto X_{\rm tr}$ is essentially surjective. This is easily proved using Hironaka: embed a given smooth $Y/\eta$ into a proper $\bar Y_0/S$ by Nagata, then resolve the singularities to get a regular $\bar Y/S$ such that $\bar Y\setminus Y$ is a divisor with normal crossings. Finally, endow $\bar Y$ with the log structure associated to the inclusion $Y\hookrightarrow \bar Y$. Then $\bar Y/S$ is log smooth (this fails if $k$ has positive characteristic). It may not be semistable, and the morphism to $S$ might not be saturated, but I guess we don't care.
Target theorem. The functor $X\mapsto X_s^{\rm log}$ factors through $X\mapsto X_{\rm tr}$ (at least if we change the target category into the homotopy category).
I will sketch how I would prove it. Let $Y$ be an object of $C$. The second functor is essentially surjective; let $X$ be an object of $\bar C$ with $X_{\rm tr} \cong Y$. We want to check that $Y\mapsto X_s^{\rm log}$ is independent of $X$. Given two such $X, X'$, we can find a third one related to both by log blow-ups $X''\to X$, $X''\to X'$. Log blow-ups are log etale and should not change the Kato-Nakayama space of the special fiber, so the induced $(X'')^{\rm log}_s\to X^{\rm log}$ are isomorphisms. For functoriality, given a map $f:Y'\to Y$ in $C$ one can using Hironaka find a `model' $X'\to X$ in $\bar C$ and look at the associated map $(X')^{\rm log}_s\to X^{\rm log}_s$.
End note. We could even work with proper formal schemes over $S$ and the theory should be the same.
Below is my initial answer. Let $X/\mathbb{C}((t))$ be a smooth proper variety. I will define the bundle over $\mathbb{S}^1$ you want up to a finite covering $\mathbb{S}^1\to\mathbb{S}^1$.
After a finite ramified base change, $X$ has a log smooth (semistable) model $\mathscr{X}/\mathbb{C}[[t]]$ where $\mathbb{C}[[t]]$ is given the natural log structure. Let $X_0/S$ be the log special fiber over the log point $S={\rm Spec}(\mathbb{N}\to\mathbb{C})$. Consider the associated map on Kato–Nakayama spaces $$ X_0^{\rm log} \to S^{\log}\cong \mathbb{S}^1. $$ This is a locally trivial bundle over $\mathbb{S}^1$ (because $X_0/S$ is log smooth and proper), and if the model $\mathscr{X}/\mathbb{C}[[t]]$ arises as the base change of an algebraic family, it agrees with what happens topologically on the generic fiber.
If $X$ is non-smooth or non-proper, you can probably use simplicial methods to reduce to the smooth proper case, adding some horizontal log structure.
P.S. If I find the time, I will try to add some more details later today.