  We consider the point process \begin{align*} \frac{1}{Z_{n}}\prod_{1 \leq j <
k \leq n} |e^{i\theta_{j}}-e^{-i\theta_{k}}|^{\beta}\prod_{j=1}^{n}
d\theta_{j}, \qquad \theta_{1},\ldots,\theta_{n} \in (-\pi,\pi], \quad \beta >
0, \end{align*} where $Z_{n}$ is the normalization constant. The feature of
this process is that the points $e^{i\theta_{1}},\ldots,e^{i\theta_{n}}$
interact with the mirror points reflected over the real line
$e^{-i\theta_{1}},\ldots,e^{-i\theta_{n}}$.
  We study smooth linear statistics of the form $\sum_{j=1}^{n}g(\theta_{j})$
as $n \to \infty$, where $g$ is $2\pi$-periodic. We prove that a wide range of
asymptotic scenarios can occur: depending on $g$, the leading order
fluctuations around the mean can (i) be of order $n$ and purely Bernoulli, (ii)
be of order $1$ and purely Gaussian, (iii) be of order $1$ and purely
Bernoulli, or (iv) be of order $1$ and of the form $BN_{1}+(1-B)N_{2}$, where
$N_{1},N_{2}$ are two independent Gaussians and $B$ is a Bernoulli that is
independent of $N_{1}$ and $N_{2}$. The above list is not exhaustive: the
fluctuations can be of order $n$, of order $1$ or $o(1)$, and other random
variables can also emerge in the limit.
  We also obtain large $n$ asymptotics for $Z_{n}$ (and some generalizations),
up to and including the term of order $1$.
  Our proof is inspired by a method developed by McKay and Wormald [10] to
estimate related $n$-fold integrals.
