  We call an element $U$ conditionally universal for a sequential convergence
space $\mathbf{\Omega}$ with respect to a minimal system
$\{\varphi_n\}_{n=1}^\infty$ in a continuously and densely embedded Banach
space $\mathcal{X}\hookrightarrow\mathbf{\Omega}$ if the partial sums of its
phase-modified Fourier series is dense in $\mathbf{\Omega}$. We will call the
element $U$ almost universal if the change of phases (signs) needs to be
performed only on a thin subset of Fourier coefficients. In this paper we prove
the existence of an almost universal element under certain assumptions on the
system $\{\varphi_n\}_{n=1}^\infty$. We will call a function $U$ asymptotically
conditionally universal in a space $L^1(\mathcal{M})$ if the partial sums of
its phase-modified Fourier series is dense in $L^1(F_m)$ for an ever-growing
sequence of subsets $F_m\subset\mathcal{M}$ with asymptotically null
complement. Here we prove the existence of such functions $U$ under certain
assumptions on the system $\{\varphi_n\}_{n=1}^\infty$. Moreover, we show that
every integrable function can be slightly modified to yield such a function
$U$.
  In particular, we establish the existence of almost universal functions for
$L^p([0,1])$, $p\in(0,1)$, and asymptotically conditionally universal functions
for $L^1([0,1])$, with respect to the trigonometric system.
