An RUC (randomly unconditionally converging) system in a Banach space \(X\) is a biorthogonal system \((x_j,x_j^\ast)\) in \(X\times X^\ast\) such that for every \(x\) in the closed linear span of the \(x_n\), the series \(\sum\limits_{j=1}^\infty r_j(t)x_j^\ast(x)x_j\) converges for almost all \(t\in[0,1]\), where \(\{r_n\}_{n=1}^\infty\) denotes the usual Rademacher sequence. Equivalently, there exists a constant \(K>0\) satisfying for all scalars \(c_1,\ldots,c_n\) \[ \displaystyle\int_0^1\|\sum\limits_{j=1}^nr_j(t)c_jx_j\|_{_X}dt\leq K\|\sum\limits_{j=1}^nc_jx_j\|_{_X} \] For example, the usual trigonometric system is an RUC system in \(L^p([0,1])\) for every \(p\geq 2\), but is an unconditional basis only if \(p=2\). It is known that every separable Banach space containing \(c_0\) admits an RUC system: see \textit{P. Billard} and \textit{C. Samuel} [C. R. Acad. Sci., Paris, Sér. I 303, 467-469 (1986; Zbl 0595.46015)]\ and \textit{P. Wojtaszczyk} [Functional analysis, Proc. 4th Annu. Sem., Austin/TX 1985-86, 37-39 (1986; Zbl 0749.46012)]. Nevertheless, the usual space \(C([0,1])\) does not have a complete RUC system which is orthonormal in \(L^2\) [see \textit{P. Billard}, Monatsh. Math. 108, 23-27 (1989; Zbl 0679.46009)]. The aim of the present paper is to prove the following characterization (Theorem 2.8): if \(E\) is a rearrangement invariant (Banach) function space on \([0,1]\), then the following assertions are equivalent. i) Each orthonormal uniformly bounded system is an RUC system in \(E\). ii) There exists a complete orthonormal uniformly bounded system which is a complete RUC system in \(E\). iii) There exists a complete orthonormal system which is weakly null and a complete RUC system in \(E\) and which is bounded in \(L^p\) for some \(p>2\). iv) The continuous embeddings \(G\subset E\subset L^2\) hold (where \(G\) is the usual gaussian Orlicz space).