Principal Components Analysis of Residuals (PCAR) is used to discover secondary dimensions in the unexplained variance after the Rasch dimension has been removed from ordinal data. Guidance is given as to when secondary dimensions are problematic and what action can be taken.
Missing data? Missing data are assigned residuals of zero when the correlations are calculated.
This video follows the method of https://www.rasch.org/rmt/rmt191h.htm but applies it to your measure distributions instead of to hypothetical measure distributions, so your simulation is more exact for your situation.
www.winsteps.com