  A tagged arc on a surface is introduced by Fomin, Shapiro, and Thurston to
study cluster theory on marked surfaces. Given a tagged arc system on a graded
marked surface, we define its $\mathbb{Z}$-graded
$\mathcal{A}_\infty$-category, generalizing the construction of Haiden,
Katzarkov, and Kontsevich for arc systems. When a tagged arc system arises from
a non-trivial involution on a marked surface, we show that this
$\mathcal{A}_\infty$-category is quasi-isomorphic to the invariant part of the
topological Fukaya category under the involution. In particular, this
identifies tagged arcs with non-geometric idempotents of Fukaya category.
