Consider the standard trigonometric system \({\mathcal T}:=\left\{(2\pi)^{-1/2}e^{ikx}\right\}_{k\in{\mathbb Z}}\) on \({\mathbb T}=[-\pi,\pi)\) in the space \(L^p({\mathbb T};w):=\left\{f:{\mathbb T}\rightarrow{\mathbb C}:\;\|f\|^p_{p,w}=\int_{-\pi}^{\pi}|f(t)|^p w(t)\, dt<\infty\right\}\), \(1<p<\infty\), where \(w\) is a nonnegative \(2\pi\)-periodic weight. In this paper, a complete characterization of \(2\pi\)-periodic weights \(w\) for which the trigonometric system \(\mathcal T\) forms a quasi-greedy basis for \(L^p({\mathbb T};w)\) is given. The characterization implies that this can happen only for \(p=2\) and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.