Inclusion Criteria:  Must have signed written, informed consent. Males or females, ≥18 years of age. Must have a confirmed diagnosis of active MM. Must have measurable MM. Must have relapsed / refractory MM, having received treatment with a proteasome inhibitor, immunomodulatory agent (IMiD), and anti-CD38 therapy. Must be willing to practice birth control from the time of Screening and throughout the first year of the study after P-BCMA-ALLO1 administration. Must have a negative serum pregnancy test at Screening and a negative urine pregnancy test within 3 days prior to initiating the lymphodepletion chemotherapy regimen (females of childbearing potential). Must be at least 90 days since autologous stem cell transplant, if performed. Must have adequate vital organ function within pre-determined parameters. Must have recovered from toxicities due to prior therapies. Must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1.  Exclusion Criteria:  Is pregnant or lactating. Has inadequate venous access. Has active hemolytic anemia, plasma cell leukemia, Waldenstrom's macroglobulinemia, POEMS syndrome, disseminated intravascular coagulation, leukostasis, or amyloidosis. Has an active second malignancy (not disease-free for at least 5 years) in addition to MM, excluding low-risk neoplasms such as non-metastatic basal cell or squamous cell skin carcinoma. Has active autoimmune disease. Has a history of significant central nervous system (CNS) disease, such as stroke, epilepsy, etc. Has an active systemic infection. Has a history of hepatitis B, hepatitis C virus, human immunodeficiency virus (HIV), or human T-lymphotropic virus (HTLV) infection, or any immunodeficiency syndrome. Has New York Heart Association (NYHA) Class III or IV heart failure, unstable angina, or a history of myocardial infarction or significant arrhythmia. Has any psychiatric or medical disorder that would preclude safe participation in and/or adherence to the protocol. Has received prior gene therapy or gene-modified cellular immunotherapy or T cell engaging antibodies. Has received anti-cancer medications within 2 weeks of the time of initiating conditioning chemotherapy. Has received immunosuppressive medications within 2 weeks of the time of administration of P-BCMA-ALLO1, and/or expected to require them while on study. Has received systemic corticosteroid therapy within 1 week or 5 half-lives (whichever is shorter) of the administration of P-BCMA-ALLO1 or is expected to require it during the course of the study. Has CNS metastases or symptomatic CNS involvement of their myeloma. Has a history of severe immediate hypersensitivity reaction to any of the agents used in this study. Has a history of having undergone allogeneic stem cell transplantation, or any other allogeneic or xenogeneic transplant, or has undergone autologous transplantation within 90 days.