Getting npsp.TDTM_Contact error upon blanking out Mailing Address fields

My client is trying to blank out an address where a Contact has moved and has left no forwarding address.

When blanking out the Mailing Address fields on Contact we get the following error:

Error: Invalid Data. 

Review all error messages below to correct your data.

Apex trigger npsp.TDTM_Contact caused an unexpected exception, contact your administrator: npsp.TDTM_Contact: execution of AfterUpdate caused by: System.DmlException: Insert failed. First exception on row 0; first error: CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY, npsp.TDTM_Address: maximum trigger depth exceeded Contact trigger event AfterUpdate for [0030W00003OQHqI] Address trigger event AfterInsert for [a2E0W000001qcmz] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI] Contact trigger event AfterUpdate for [0030W00003OQHqI]: []: (npsp)

We can save fine if replacing the old address with a new address.

Address Override is not checked.

I cannot see any workflow that would impact the address either on Contact or Address.

I also tried deleting the Address record itself. This did not cause an error, but the old address was still showing on the Contact record. When I tried to delete the address on Contact after deleting the Address record, I got the same error again.

I tried blanking out the address through the "Manage Household" interface and got the same error.

@Nonprofit Success Pack