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.
• Is positive for cytomegalovirus (CMV) by PCR, CMV immunoglobulin M (IgM) antibody, or Coronavirus disease 2019 (COVID-19) by PCR.
• 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 allogeneic cellular therapy or gene therapy.
• 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.
• Arms R, RS, RP1, RP1.5 and RP2 Only: a) Has received a live vaccine within the last 28 days of the first administration of agents used in Arm R or RS, b) Has any known hypersensitivity or severe reactions or toxicity to agents used in Arms R or RS.