  Hall algebras and related constructions have had diverse applications in
mathematics and physics, ranging from representation theory and quantum groups
to Donaldson-Thomas theory and the algebra of BPS states. The theory of
$2$-Segal spaces was introduced independently by Dyckerhoff-Kapranov and
G\'alvez-Carrillo-Kock-Tonks as a unifying framework for Hall algebras: every
$2$-Space defines an algebra in the $\infty$-category of spans, and different
Hall algebras correspond to different linearisations of this universal Hall
algebra.
  A recurring theme is that Hall algebras can often be equipped with a
coproduct which makes them a bialgebra, possibly up to a `twist'. In this paper
will explain the appearance of these bialgebraic structures using the theory of
$2$-Segal spaces: We construct the universal Hall bialgebra of a double
$2$-Segal space, which is a lax bialgebra in the $(\infty,2)$-category of
bispans. Moreover, we show how examples of double $2$-Segal spaces arise from
Waldhausen's $S$-construction.
