  We give two constructions of functorial topological realizations for schemes
of finite type over the field $\mathbb{C}(\!(t)\!)$ of formal Laurent series
with complex coefficients, with values in the homotopy category of spaces over
the circle. The problem of constructing such a realization was stated by D.
Treumann, motivated by certain questions in mirror symmetry. The first
construction uses spreading out and the usual Betti realization over
$\mathbb{C}$. The second uses generalized semistable models and log Betti
realization defined by Kato and Nakayama, and applies to smooth rigid analytic
spaces as well. We provide comparison theorems between the two constructions
and relate them to the \'etale homotopy type and de Rham cohomology. As an
illustration of the second construction, we treat two examples, the Tate curve
and the non-archimedean Hopf surface.
