  For numerous online bipartite matching problems, such as edge-weighted
matching and matching under two-sided vertex arrivals, the state-of-the-art
fractional algorithms outperform their randomized integral counterparts. This
gap is surprising, given that the bipartite fractional matching polytope is
integral, and so lossless rounding is possible. This gap was explained by
Devanur et al.~(SODA'13), who showed that \emph{online} lossless rounding is
impossible.
  Despite the above, we initiate the study of lossless online rounding for
online bipartite matching problems. Our key observation is that while lossless
online rounding is impossible \emph{in general}, randomized algorithms induce
fractional algorithms of the same competitive ratio which by definition are
losslessly roundable online. This motivates the addition of constraints that
decrease the ``online integrality gap'', thus allowing for lossless online
rounding. We characterize a set of non-convex constraints which allow for such
lossless online rounding, and better competitive ratios than yielded by
deterministic algorithms.
  As applications of our lossless online rounding approach, we obtain two
results of independent interest: (i) a doubly-exponential improvement, and a
sharp threshold for the amount of randomness (or advice) needed to outperform
deterministic online (vertex-weighted) bipartite matching algorithms, and (ii)
an optimal semi-OCS, matching a recent result of Gao et al.~(FOCS'21) answering
a question of Fahrbach et al.~(FOCS'20).
