  A subvariety of a quasi-projective complex variety $X$ is called
``universally irreducible'' if its preimage inside the universal cover of $X$
is irreducible. In this paper we investigate sufficient conditions for
universal irreducibility. We consider in detail complete intersection
subvarieties of small codimension inside Siegel moduli spaces of any finite
level. Moreover we show that, for $g\geq 3$, every Siegel modular form is the
product of finitely many irreducible analytic functions on the Siegel upper
half-space $\mathbb{H}_g$. We also discuss the special case of singular theta
series of weight $\frac{1}{2}$ and of Schottky forms.
