  For $1<p<\infty$, we emulate the Bergman projection on Reinhardt domains by
using a Banach-space basis of $L^p$-Bergman space. The construction gives an
integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined
by the kernel is shown to be absolutely bounded projection on the $L^p$-Bergman
space on a class of domains where the $L^p$-boundedness of the Bergman
projection fails for certain $p \neq 2$. As an application, we identify the
duals of these $L^p$-Bergman spaces with weighted Bergman spaces.
