A Study of MCARH109 and MCARH125 in People With Multiple Myeloma

Inclusion Criteria: * Patients must have histologically confirmed multiple myeloma (MM) by MSKCC pathologist. * Age ≥ 18 years of age * Diagnosis of relapsed or refractory MM with at least 3 prior lines of therapy. * Refractory myeloma is defined as disease that progresses while on therapy or within 60 days after the last therapy. Relapsed myeloma is defined as previously treated myeloma with initial response and subsequent progression (per International Myeloma Working Group i.e. IMWG criteria) not meeting criteria for refractory disease. * At least 3 prior lines of therapy; Prior therapy should include all of the following: 1. A proteasome inhibitor (e.g., bortezomib, carfilzomib, ixazomib) 2. An immunomodulatory drug (e.g., thalidomide, lenalidomide, pomalidomide) 3. A CD38 monoclonal antibody (e.g., daratumumab) 4. High dose chemotherapy with autologous stem cell support (ASCT) * Subjects who are not candidates to receive one or more of the above treatments (5 a-d) are eligible for the trial. * ECOG performance status of 0 or 1 * HGB ≥ 8 g/dl, ANC≥ 1,000/mm3, Platelet≥ 50,000/mm3 without red cell transfusion for 21 days, platelet transfusion for 7 days and or growth factor support (Neupogen or Neulasta) for at least 14 days prior to initial screening (screening A). HGB ≥ 8 g/dl, ANC ≥ 1,000/mm3, Platelet ≥ 20,000/mm3 prior to pre-treatment screening (screening B). Patients are allowed to receive transfusion support prior to the pre-treatment screening but no growth factor support (Neupogen or Neulasta) for 7 days prior to pre-treatment screening. * Measurable disease defined as meeting at least one of the criteria below: 1. Serum M protein ≥ 0.5 g/dL 2. Involved serum free light chain ≥10 mg/dL with an abnormal free light chain ratio 3. Urine M-protein ≥ 200 mg/24 hours 4. Measurable plasmacytomas seen on imaging (≥ 1 lesion that has a single diameter ≥ 2 cm). If this is the primary marker of measurable disease, patients will need a biopsy at the pre-treatment screening (screening B). 5. Bone marrow plasma cells ≥ 30% as determined by CD138 immunohistochemistry staining * Serum creatinine ≤ 1.5mg/dL or a measured creatinine clearance ≥ 50 mL/min (using 24-hour urine collection) * Alanine Aminotransferase (ALT) and Aspartate Aminotransferase (AST) ≤ 3 X ULN and total bilirubin ≤ 2 mg/dL (or \< 3 mg/dL for individuals with Gilbert's syndrome) * PT and PTT ≤ 1.5 X ULN * Adequate pulmonary function as assessed by ≥92% oxygen saturation on room air by pulse oximetry * Adequate cardiac function, defined as LVEF ≥ 40% by transthoracic echocardiogram (TTE) or multigated acquisition scan (MUGA) performed within 4 weeks of initial screening * For patients with prior ASCT, at least 100 days since ASCT at the time of initial screening. Exclusion Criteria: * Pregnant or lactating women. Women and men of childbearing age should use highly effective contraception while on this study and continue for 1 year after all treatment is finished. * Patients with following cardiac conditions will be excluded: * New York Heart Association (NYHA) stage III or IV congestive heart failure * Myocardial infarction ≤6 months prior to enrollment * History of clinically significant ventricular arrhythmia or unexplained syncope, not believed to be vasovagal in nature or due to dehydration * History of severe non-ischemic cardiomyopathy * Patients with HIV or active hepatitis B or hepatitis C infection are ineligible. * Current diagnosis of primary and secondary plasma cell leukemia is excluded. History of plasma cell leukemia is not excluded. * Patients who have not received any myeloma therapy for the preceding 6 months except if the last myeloma therapy was a CAR T cell therapy. * At least 14 day washout from myeloma therapies prior to leukapheresis and prior to starting lymphodepletion. The washout for experimental treatments would be 5 half-lives or 14 days (whichever is shorter). * At least 14 day washout from radiation prior to leukapheresis and prior to starting lymphodepletion. * Patients treated with previous JCARH125, and/or MCARH109 (any other cellular products that have the same construct) will be excluded. Other CART therapies are not excluded. Prior therapies with other BCMA or GPRC5D targeted therapies including antibody drug conjugates and bispecific antibodies are not excluded. * Patients with any concurrent active malignancies (or another primary malignancy not in remission for at least 2 years) as defined by malignancies requiring any therapy other than expectant observation or hormonal therapy, with the exception of squamous and basal cell carcinoma of skin. * Patients with a prior allogeneic transplant at least 6 months prior to study enrollment are eligible unless experienced GvHD that required systemic steroids or other systemic lymphotoxic therapy within 12 weeks of initial screening * Patients on systemic steroids (except if solely for adrenal replacement) within two weeks of collection * Active auto-immune disease including connective tissue disease, uveitis, sarcoidosis, inflammatory bowel disease, or multiple sclerosis, or have a history of severe (as judged by the principal investigator) autoimmune disease requiring prolonged immunosuppressive therapy * Prior or active CNS involvement by myeloma (e.g., leptomeningeal disease). Screening for this, for example, by lumbar puncture, is only required if suspicious symptoms or radiographic findings are present. * Pre-existing (active or severe) neurologic disorders (e.g., pre-existing seizure disorder) * Active uncontrolled acute infections * Any other issue which, in the opinion of the treating physician, would make the patient ineligible for the study.