  We study inverse source problems associated to semilinear elliptic equations
of the form
  \[
  \Delta u(x)+a(x,u)=F(x),
  \]
  on a bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq 2$. We show that it
is possible to use nonlinearity to break the gauge symmetry of the inverse
source problem for a class of nonlinearities $a(x,u)$. This is in contrast to
inverse source problems for linear equations, which always have a gauge
symmetry. The class of nonlinearities include certain polynomials and
exponential nonlinearities. For these nonlinearities, we determine both
$a(x,u)$ and $F(x)$ uniquely from the associated DN map.
  Moreover, for general nonlinearities $a(x,u)$, we show that we can recover
the derivatives $\partial_u^ka(x,u)$ and the source $F(x)$ up to a gauge.
Especially, we recover general polynomial nonlinearities up to a gauge and
generalize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$
is a solution.
