Verbatim · Ayatollah A. Sistani
Sitting in the courtyards and outer parts of the building of the shrine of the infallible Imams which are not designated as a mosque (Masjid) is permissible. However in the building in which the shrine of the infallible Imam (as) is located, based on obligatory precaution, she is not allowed to sit there but she is allowed to enter from a door and exit from another.
Ayatollah A. Sistani sistani.org Q&A #26335 other marjas may differ
A. Khamenei muqallids: per Ajwibat al-Istiftāʾāt Q221, entering the shrines of the Imams’ descendants (sayyid shrines other than the Imam mausolea themselves) is permissible. For the Imam shrines themselves, A. Khamenei rules separately (Q422) that entry is not permissible for a menstruating woman or a junub person under the blessed dome and the area commonly recognised as the shrine, a stricter framing than A. Sistani’s “obligatory precaution” wording above.