your proposed patch is very very specific to DTI/TDI conversions. Please try to narrow down exactly what is going wrong here, this is introducing a massive level of complexity.

The bug is very specific to DTI/TDI conversions.

What I did here is reproduce what happen when joining a DatetimeLike index with another index, see code here

The reason this bug happen it that join between DatetimeLike Index and Object Index works, ie, the returned join_index has casted the object to the corresponding datetimelike, but then ,when calling get_indexer,

In brief, in Index.join, for multiindex, we join datetimelike and object together with maybe_casting of the object, but this doesn't happen at the get_indexer level.
When using single Index, the object index is maybe_casted before attempting to join, and the get_indexer then work.

woa, we don't want to do anything like this.

That is what happen actually for joining DatetimeLikeIndex with ObjectIndex

I am open to suggestion if one could tell me what we expect here.
The actual patch try to make the comportement of multiindex join consistent with the one on single index.
An alternativ is to prevent joining datetimelike with object, and raise by asking to explicitly cast to datetimelike before end, but I suspect this would be a breaking change.

@nrebena if indices type are not the same then they must be joined as object

@jreback This is actually not the case for single index, see this example, where the indice type differs, but the resulting type is DatetimeIndex.

import pandas as pd
import datetime

i1 = pd.Index(['2019-05-31'])
i2 = pd.Index([datetime.datetime(2019, 5, 31)])

print(i1.join(i2, return_indexers=True))
# (DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None), None, None)

Is it an acceptable solution to fix Index.join to return an object index in this case ?

yes see what fixing this breaks (likely would need to depreciate this)

Update on root cause and advice needed.
In the following example we have one datetime index and one string index with a datetime, with their comportment regarding join, intersection, union and equals.
I am not sure what are the desired comportment, but actually, anytime a string can be use as a datetime, it is casted so. See the following result from branch master.

import pandas as pd
import datetime

i1 = pd.Index(['2019-05-31'])
i2 = pd.Index([datetime.datetime(2019, 5, 31)])

i2.join(i1, how='outer', return_indexers=True))
# (DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None), None, None)
i1.join(i2, how='outer', return_indexers=True) 
# (DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None), None, None)

i1.union(i2)
# Index(['2019-05-31', 2019-05-31 00:00:00], dtype='object')
i2.union(i1)                                                                                                 
# Index([2019-05-31 00:00:00, '2019-05-31'], dtype='object')

i1.intersection(i2)
# Index(['2019-05-31'], dtype='object')
i2.intersection(i1)                                                                                          
# DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None)

i1.equals(i2)
# True
i2.equals(i1)                                                                                                
# True

date = "2020-03-16"
values = [date, pd.to_datetime(date)]
pd.Categorical(values, ordered=False)
# [2020-03-16, 2020-03-16]
# Categories (1, datetime64[ns]): [2020-03-16]

Are this results what should be expected for each one? For the one with unexpected result, what should be the right one?

Update on root cause and advice needed.
In the following example we have one datetime index and one string index with a datetime, with their comportment regarding join, intersection, union and equals.
I am not sure what are the desired comportment, but actually, anytime a string can be use as a datetime, it is casted so. See the following result from branch master.

import pandas as pd
import datetime

i1 = pd.Index(['2019-05-31'])
i2 = pd.Index([datetime.datetime(2019, 5, 31)])

i2.join(i1, how='outer', return_indexers=True))
# (DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None), None, None)
i1.join(i2, how='outer', return_indexers=True) 
# (DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None), None, None)

i1.union(i2)
# Index(['2019-05-31', 2019-05-31 00:00:00], dtype='object')
i2.union(i1)                                                                                                 
# Index([2019-05-31 00:00:00, '2019-05-31'], dtype='object')

i1.intersection(i2)
# Index(['2019-05-31'], dtype='object')
i2.intersection(i1)                                                                                          
# DatetimeIndex(['2019-05-31'], dtype='datetime64[ns]', freq=None)

i1.equals(i2)
# True
i2.equals(i1)                                                                                                
# True

date = "2020-03-16"
values = [date, pd.to_datetime(date)]
pd.Categorical(values, ordered=False)
# [2020-03-16, 2020-03-16]
# Categories (1, datetime64[ns]): [2020-03-16]

Are this results what should be expected for each one? For the one with unexpected result, what should be the right one?

all of the above should be deprecated. We shouldn't every compare strings and datetimes. We may currently do this, but we need to deprecate it.

@nrebena if you would like to update this to the above behavior would be fine; we should not be joining datetimelike things with non-datetimelike things at all.

Sure thing. I'll try to get to it.

I decided to split this, and started another PR here #33531, and will continue this one regarding the behaviour of pd.Categorical.

closing this until / unless #33531 is resolved