  We extend the real-space renormalization group (RG) approach to the study of
the energy level statistics at the integer quantum Hall (QH) transition.
Previously it was demonstrated that the RG approach reproduces the critical
distribution of the {\em power} transmission coefficients, i.e., two-terminal
conductances, $P_{\text c}(G)$, with very high accuracy. The RG flow of $P(G)$
at energies away from the transition yielded the value of the critical
exponent, $\nu$, that agreed with most accurate large-size lattice simulations.
To obtain the information about the level statistics from the RG approach, we
analyze the evolution of the distribution of {\em phases} of the {\em
amplitude} transmission coefficient upon a step of the RG transformation. From
the fixed point of this transformation we extract the critical level spacing
distribution (LSD). This distribution is close, but distinctively different
from the earlier large-scale simulations. We find that away from the transition
the LSD crosses over towards the Poisson distribution. Studying the change of
the LSD around the QH transition, we check that it indeed obeys scaling
behavior. This enables us to use the alternative approach to extracting the
critical exponent, based on the LSD, and to find $\nu=2.37\pm0.02$ very close
to the value established in the literature. This provides additional evidence
for the surprising fact that a small RG unit, containing only five nodes,
accurately captures most of the correlations responsible for the
localization-delocalization transition.
