A function on a compact (complex) Kähler manifold $X$ is quasi-psh if it is locally the sum of a psh function and a smooth function. Given a closed $(1,1)$-form $\theta$, an $\theta$-psh function $\varphi$ is a quasi-psh function such that $\theta+dd^{c}\varphi\geq 0$ in the sense of currents. When the de Rham class $\{\theta\}\in H^{1,1}(X)$ is a Kähler class, we have a global characterization: $\theta$-psh functions are decreasing limits of sequences of smooth $\theta$-psh functions, see [Dem92, Theorem 1.1]. In our current non-Archimedean setting, a local theory of psh functions is still to be developed. For this reason, we work globally and assume that $\{\theta\}$ is ample.