  The census of open clusters has exploded in size thanks to data from the Gaia
satellite. However, it is likely that many of these reported clusters are not
gravitationally bound, making the open cluster census impractical for many
scientific applications. We test different physically motivated methods for
distinguishing between bound and unbound clusters, using them to create a
cleaned cluster catalogue. We derived completeness-corrected photometric masses
for 6956 clusters from our earlier work. Then, we used these masses to compute
the size of the Roche surface of these clusters (their Jacobi radius) and
distinguish between bound and unbound clusters. We find that only 5647 (79%) of
the clusters from our previous catalogue are compatible with bound open
clusters, dropping to just 11% of clusters within 250 pc. 3530 open clusters
are in a strongly cut high quality sample. The moving groups in our sample show
different trends in their size as a function of age and mass, suggesting that
they are unbound and undergoing different dynamical processes. Our cluster mass
measurements constitute the largest catalogue of Milky Way cluster masses to
date, which we also use for further science. Firstly, we inferred the
mass-dependent completeness limit of the open cluster census, showing that the
census is complete within 1.8 kpc only for objects heavier than 230 M$_\odot$.
Next, we derived a completeness-corrected age and mass function for our open
cluster catalogue, including estimating that the Milky Way contains a total of
$1.3 \times 10^5$ open clusters, only ~4% of which are currently known.
Finally, we show that most open clusters have mass functions compatible with
the Kroupa initial mass function. We demonstrate Jacobi radii for
distinguishing between bound and unbound star clusters, and publish an updated
star cluster catalogue with masses and improved cluster classifications.
(abridged)
