  A half-integral polygon with quasi-period collapse behaves similarly to a
lattice polygon in the sense that the number of lattice points in its integer
dilates can be calculated as values of a polynomial, its Ehrhart polynomial. As
a main result, we classify the Ehrhart polynomials of all half-integral
non-lattice polygons with quasi-period collapse. In particular, we obtain that
for any positive integer $i$, the polynomial
$\frac{4i+5}{2}t^2+\frac{2i+7}{2}t+1\in \mathbb{Q}[t]$ is an Ehrhart polynomial
of a rational polygon, which was an open question for $i>1$.
  We also study some extreme cases in detail. In particular, we show that up to
affine unimodular equivalence there exist exactly $30$ half-integral
non-lattice polygons with quasi-periodic collapse with exactly one interior
lattice point, which are the dual polygons of the $30$ LDP polygons of
Gorenstein index $2$. Furthermore, we classify all half-integral polygons with
quasi-period collapse with at most $6$ interior lattice points or with $i\geq
1$ interior lattice points and the maximum possible number $2i+7$ of boundary
lattice points.
