BUG: Regression of Index.join() when return_indexers=True

[michaelpradel]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd
index1 = pd.Index([1, 3, 4])
index2 = pd.Index([2, 2, 3])
joined_index, lidx, ridx = index1.join(index2, how='right', return_indexers=True)
print(joined_index, lidx, ridx)

Issue Description

Unlike Pandas 2.2.2, the current main branch doesn't always return the indexers. The above example prints this output (where the right indexer surprisingly is None):
Index([2, 2, 3], dtype='int64') [-1 -1 1] None

The change in behavior got introduced with #56841, which is supposed to improve performance, but not to change the behavior in other ways.

Expected Behavior

Pandas 2.2.2 prints this output, which is what I'd expect:
Index([2, 2, 3], dtype='int64') [-1 -1 1] [0 1 2]

Installed Versions

INSTALLED VERSIONS ------------------ commit : 9fee6cf python : 3.10.8.final.0 python-bits : 64 OS : Linux OS-release : 6.5.0-28-generic Version : #29-Ubuntu SMP PREEMPT_DYNAMIC Thu Mar 28 23:46:48 UTC 2024 machine : x86_64 processor : byteorder : little LC_ALL : None LANG : C.UTF-8 LOCALE : en_US.UTF-8

pandas : 3.0.0.dev0+850.g9fee6cfc59
numpy : 1.26.4
pytz : 2024.1
dateutil : 2.9.0.post0
setuptools : 63.2.0
pip : 24.0
Cython : 3.0.10
pytest : 8.2.0
hypothesis : 6.100.2
sphinx : 7.3.7
blosc : None
feather : None
xlsxwriter : 3.2.0
lxml.etree : 5.2.1
html5lib : 1.1
pymysql : 1.4.6
psycopg2 : 2.9.9
jinja2 : 3.1.3
IPython : 8.24.0
pandas_datareader : None
adbc-driver-postgresql: None
adbc-driver-sqlite : None
bs4 : 4.12.3
bottleneck : 1.3.8
fastparquet : 2024.2.0
fsspec : 2024.3.1
gcsfs : 2024.3.1
matplotlib : 3.8.4
numba : 0.59.1
numexpr : 2.10.0
odfpy : None
openpyxl : 3.1.2
pyarrow : 16.0.0
pyreadstat : 1.2.7
python-calamine : None
pyxlsb : 1.0.10
s3fs : 2024.3.1
scipy : 1.13.0
sqlalchemy : 2.0.29
tables : 3.9.2
tabulate : 0.9.0
xarray : 2024.3.0
xlrd : 2.0.1
zstandard : 0.22.0
tzdata : 2024.1
qtpy : None
pyqt5 : None