  Recent gravitational-wave transient catalogs have used \pastro{}, the
probability that a gravitational-wave candidate is astrophysical, to select
interesting candidates for further analysis. Unlike false alarm rates, which
exclusively capture the statistics of the instrumental noise triggers,
\pastro{} incorporates the rate at which triggers are generated by both
astrophysical signals and instrumental noise in estimating the probability that
a candidate is astrophysical. Multiple search pipelines can independently
calculate \pastro{}, each employing a specific data reduction. While the range
of \pastro{} results can help indicate the range of uncertainties in its
calculation, it complicates interpretation and subsequent analyses. We develop
a statistical formalism to calculate a \emph{unified} \pastro{} for
gravitational-wave candidates, consistently accounting for triggers from all
pipelines, thereby incorporating extra information about a signal that is not
available with any one single pipeline. We demonstrate the properties of this
method using a toy model and by application to the publicly available list of
gravitational-wave candidates from the first half of the third LIGO-Virgo-KAGRA
observing run. Adopting a unified \pastro{} for future catalogs would provide a
simple and easy-to-interpret selection criterion that incorporates a more
complete understanding of the strengths of the different search pipelines
