Say a flow $\Psi$ on the $n$-torus $\mathbb{T}^n$ is quasiperiodic if it is of the form $\Psi^t(x_1,\ldots,x_n) = (x_1 + \omega_1 t \mod 2\pi, \ldots, x_n + \omega_n t \mod 2\pi)$, where the $\omega_i$ are linearly independent over $\mathbb{Z}$.
Let $\Phi$ be a continuous flow on a compact metric space $X$. Say that $\Phi$ has an $n$-dimensional continuous quasiperiodic factor (i.e., semiconjugacy) if there is a continuous map $f:X\to \mathbb{T}^n$ and quasiperiodic flow $\Psi$ on $\mathbb{T}^n$ with
$\Psi^t \circ f = f \circ \Phi^t$
for all $t \in \mathbb{R}$.
Question: is there an example where $\Phi$ admits a continuous, surjective, $n$-dimensional quasiperiodic factor $f$ for $n \geq 2$ which is not a fiber bundle? If not, what about an example which is not a (Hurewicz or Serre) fibration? What about an example in the smooth category?