  We continue our investigation of contractive projections on noncommutative
$\mathrm{L}^p$-spaces where $1 < p < \infty$ started in \cite{ArR19}. We
improve the results of \cite{ArR19} and we characterize precisely the positive
contractive projections on a noncommutative $\mathrm{L}^p$-space associated
with a $\sigma$-finite von Neumann algebra. We connect this topic to the theory
of $\mathrm{JW}^*$-algebras. More precisely, in large cases, we are able to
show that the range of a positive contractive projection is isometric to a
nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JW}^*$-algebra.
