We address questions on the existence and structure of universal functions for classes $L^p[0,1{)}^2$, $p\in (0,1)$, with respect to the double Walsh system. It is shown that there exists a measurable set $E\subset [0,1{)}^2$ with measure arbitrarily close to $1$, such that, by a proper modification of any integrable function $f\in L^1[0,1{)}^2$ outside $E$, we can get an integrable function $\tilde{f}\in L^1[0,1{)}^2$, which is universal for each class $L^p[0,1{)}^2$, $p\in (0,1)$, with respect to the double Walsh system in the sense of signs of Fourier coefficients.