  We further investigate the relations between the large degree asymptotics of
the number of real zeros of random trigonometric polynomials with dependent
coefficients and the underlying correlation function. We consider trigonometric
polynomials of the form \[ f_n(t):= \frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_k
\cos(kt)+b_k\sin(kt), ~x\in [0,2\pi], \] where the sequences $(a_k)_{k\geq 1}$
and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian
process centered with variance one and correlation function $\rho$ with
associated spectral measure $\mu_{\rho}$. We focus here on the case where
$\mu_{\rho}$ is not purely singular and we denote by $\psi_{\rho}$ its density
component with respect to the Lebesgue measure $\lambda$. Quite surprisingly,
we show that the asymptotics of the number of real zeros
$\mathcal{N}(f_n,[0,2\pi])$ of $f_n$ in $[0,2\pi]$ is not related to the decay
of the correlation function $\rho$ but instead to the Lebesgue measure of the
vanishing locus of $\psi_{\rho}$. Namely, assuming that $\psi_{\rho}$ is
$\mathcal{C}^1$ with H\"older derivative on an open set of full measure, one
establishes that \[ \lim_{n \to +\infty} \frac{\mathbb
E\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}=
\frac{\lambda(\{\psi_{\rho}=0\})}{\pi \sqrt{2}} + \frac{2\pi -
\lambda(\{\psi_{\rho}=0\})}{\pi\sqrt{3}}. \] On the other hand, assuming a sole
log-integrability condition on $\psi_{\rho}$, which implies that it is positive
almost everywhere, we recover the asymptotics of the independent case, i.e. the
limit is $\frac{2}{\sqrt{3}}$. Besides, with further assumptions of regularity
and existence of negative moment for $\psi_{\rho}$, we moreover show that the
above convergence in expectation can be strengthened to an almost sure
convergence.
