Testing the Investigational Medication Combination of Teclistamab and Pomalidomide Compared to the Usual Treatment (Carfilzomib, Pomalidomide, and Dexamethasone) for Patients With Multiple Myeloma Who Have Relapsed Shortly After Treatment

Inclusion Criteria: * Patient must be ≥ 18 years of age * Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Patient must have an identifiable dominant sequence (clonotype) established based on Adaptive Biotechnologies clonoSEQ® assay for Phase II only * Patient must have multiple myeloma and meet both of the following criteria for the original diagnosis of myeloma following the International Myeloma Working Group (IMWG) myeloma diagnostic criteria: * Bone marrow plasmacytosis with ≥ 10% plasma cells or sheets of plasma cells or biopsy proven plasmacytoma. * Tissue biopsy of any bone lesion or extramedullary plasmacytoma if applicable (positive/negative for clonal plasma cells) * Any one or more of the following symptomatic myeloma-defining events that prompted initiation of therapy as well as end-organ damage: * Anemia (hemoglobin value of \> 2 g/dL below the lower limit of normal, or a hemoglobin value \< 10 g/dL) * Hypercalcemia (serum calcium \> 1 mg/dL higher than the upper limit of normal or \> 11 mg/dL) * Bone disease (one or more osteolytic lesions on skeletal radiography, CT, or FDG-PET/CT) * Renal dysfunction (creatinine clearance \< 40 mL/min or serum creatinine \> 2 mg/dL). * Clonal bone marrow plasma cells (BMPCs) ≥ 60% * Involved/uninvolved serum free light chain ratio ≥ 100 * \> 1 focal lesions on magnetic resonance imaging (MRI) studies ≥ 5 mm * NOTE: Patients with smoldering myeloma (serum m protein ≥ 3 gm/dL or bone marrow plasma cells ≥ 10% or greater plus no evidence of anemia, hypercalcemia, lytic bone lesions or renal dysfunction) and monoclonal gammopathy of undetermined significance (serum m protein \< 3 gm/dL and bone marrow plasma cells \< 10% or greater plus no evidence of anemia, hypercalcemia, lytic bone lesions or renal dysfunction) are not eligible * Patient must have experienced relapse disease defined as relapse after one line of lenalidomide therapy including single agent lenalidomide or lenalidomide/dexamethasone or CD38 antibody/lenalidomide +/- dexamethasone. The relapse can be progressive disease as determined by the treating clinician and meeting measurable disease criteria * Patient must have received no more than two lines of IMWG defined therapy (induction +/- transplant + maintenance is considered a single line of therapy) * Patient must not receive non-protocol concurrent chemotherapy for the study disease, or any ancillary therapy considered investigational while participating in the study * Patient must have received an immunomodulatory drug (lenalidomide) and a CD38 antibody (daratumumab or isatuximab) * Patient must have measurable disease as defined by having one or more of the following, obtained within 28 days prior to registration/randomization: * ≥ 0.5 g/dL monoclonal protein (M-protein) on serum protein electrophoresis * ≥ 200 mg/24 hrs of monoclonal protein (M-protein) on a 24-hour urine protein electrophoresis * Involved free light chain ≥ 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio (\< 0.26 or \> 1.65) * Exception: patients without measurable disease in the serum or urine, but with bone or soft tissue plasmacytoma(s) ≥ 2 cm or who have bone marrow plasma cells ≥ 30% are eligible on study * NOTE: Urine protein electrophoresis (UPEP) (on a 24 hour collection) is required, no substitute method is acceptable. Urine must be followed monthly if the baseline urine M-spike is ≥ 200 mg/24 hr. Please note that if both serum and urine m-components are present, both must be followed in order to evaluate response * Patient must not be known to be refractory to carfilzomib or pomalidomide * Patient must not have received a prior BCMA therapy * Patient must not be pregnant due to the potential harm to an unborn fetus with the treatment regimens being used * All patients of childbearing potential must have a blood test or urine study within 14 days prior to registration/randomization to rule out pregnancy. Patients must also agree to a second pregnancy test completed within 24 hours prior to the first dose of pomalidomide and teclistamab and agree to ongoing pregnancy testing while on protocol treatment * A patient of childbearing potential is defined as anyone, regardless of whether they have undergone tubal ligation, who meets the following criteria: * Has achieved menarche at some point * Has not undergone a hysterectomy or bilateral oophorectomy * Has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Patient must not expect to conceive or father children by using accepted and effective method(s) of contraception or by abstaining from sexual intercourse for the duration of their participation in the study and for 6 months after the last dose of protocol treatment for patients on Arms A, B, and C and for 1 month after the last dose of protocol treatment for patients on Arm D. In addition, all patients must also agree to register to the mandatory Risk Evaluation and Mitigation Strategies (REMS)® program and be willing and able to comply with the requirements of the REMS® program * Patient must agree to not breastfeed while on protocol treatment due to the potential risk for adverse events in nursing infants, and must agree to continue not breastfeeding for an additional 5 months after the last dose of Arm A, B, or C protocol treatment and for an additional 28 days after the last dose of Arm D protocol treatment * Patient must agree to abstain for donating eggs (ova, oocytes) while on study treatment and for 6 months after the last dose of Arm A, B or C protocol treatment * Patient must agree to abstain from donating sperm while on study treatment and for 3 months after the last dose of protocol treatment even if they have had a successful vasectomy * Patient must agree to abstain from donating blood during study participation and for at least 28 days after the last dose of protocol treatment * Patient must not receive non-protocol concurrent chemotherapy, or any ancillary therapy considered investigational while participating in the study * NOTE: Bisphosphonates are considered supportive care rather than therapy and are thus allowed while on protocol treatment * Patient must not have peripheral neuropathy \> grade 2 on clinical examination at the time of registration/randomization * Patient must not have known uncontrolled congestive heart failure (CHF), New York Heart Association (NYHA) class III or IV, uncontrolled hypertension as defined as systolic \> 180 and/or diastolic \> 100, uncontrolled atrial fibrillation (A fib), or left ventricular ejection fraction (LVEF) \< 40% within 6 months prior to registration/randomization * Patient must not have known allergies, hypersensitivity, or intolerance to corticosteroids, monoclonal antibodies or human proteins, or their excipients, or known sensitivity to mammalian-derived products * Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible * Absolute neutrophil count (ANC) ≥ 1,500/uL (≤ 14 days prior to registration/randomization) * Platelets ≥ 100,000/uL (≤ 14 days prior to registration/randomization) * Total bilirubin ≤ 3 x institutional upper limit of normal (ULN) (≤ 14 days prior to registration/randomization) * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3.0 x institutional ULN (≤ 14 days prior to registration/randomization) * Calculated creatinine clearance ≥ 30 mL/min (≤ 14 days prior to registration/randomization) * NOTE: Cockcroft-Gault equation should be used to calculate creatinine clearance * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of registration/randomization are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with past central nervous system (CNS) involvement are eligible if there is no current CNS involvement * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial