  We introduce toric $b$-divisors on complete smooth toric varieties and a
notion of integrability of such divisors. We show that under some positivity
assumptions toric $b$-divisors are integrable and that their degree is given as
the volume of a convex set. Moreover, we show that the dimension of the space
of global sections of a nef toric $b$-divisor is equal to the number of lattice
points in this convex set and we give a Hilbert--Samuel type formula for its
asymptotic growth. This generalizes classical results for classical toric
divisors on toric varieties. Finally, we relate convex bodies associated to
$b$-divisors with Newton--Okounkov bodies. The main motivation for studying
toric $b$-divisors is that they locally encode the singularities of the
invariant metric on an automorphic line bundle over a toroidal compactification
of a mixed Shimura variety of non-compact type.
