  The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity
is a rational homology ball $B_{p,q}$. For a canonically polarised surface of
general type $X$, it is known that there are bounds on the number $p$ for which
$B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a
recipe to construct unbounded sequences of symplectically embedded $B_{p,q}$
into surfaces of general type equipped with non-canonical symplectic forms.
Ultimately, these symplectic embeddings come from Mori's theory of flips, but
we give an interpretation in terms of almost toric structures and mutations of
polygons. The key point is that a flip of surfaces, as studied by Hacking,
Tevelev and Urz\'ua, can be formulated as a combination of mutations of an
almost toric structure and deformation of the symplectic form.
