  We express the recently introduced real-time diagrammatic Quantum Monte
Carlo, Phys. Rev. B 91, 245154 (2015), in the Larkin-Ovchinnikov basis in
Keldysh space. Based on a perturbation expansion in the local interaction $U$,
the special form of the interaction vertex allows to write diagrammatic rules
in which vacuum Feynman diagrams directly vanish. This reproduces the main
property of the previous algorithm, without the cost of the exponential sum
over Keldysh indices. In an importance sampling procedure, this implies that
only interaction times in the vicinity of the measurement time contribute. Such
an algorithm can then directly address the long-time limit needed in the study
of steady states in out-of-equilibrium systems. We then implement and discuss
different variants of Monte Carlo algorithms in the Larkin-Ovchinnikov basis. A
sign problem reappears, showing that the cancellation of vacuum diagrams has no
direct impact on it.
