Genetically Engineered Cells (MUC1-Activated T-Cells) for the Treatment of MUC1 Positive Recurrent or Refractory Multiple Myeloma

Inclusion Criteria:

* Age \>= 18 years
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1
* MUC1 expression in multiple myeloma tumor cells verified by immunohistochemistry (IHC) or flow cytometry (FCM). Heterogeneous tumor expression of MUC1 is acceptable
* Relapsed or refractory multiple myeloma previously treated with or intolerant to at least three prior lines of therapy and be relapsed or intolerant to a proteasome inhibitor, an immune modulatory drug (IMiD), and a CD38 antibody. Patients must be at least 90 days since an autologous stem cell transplant, if performed
* Patients must have measurable disease per IMWG criteria on study entry, which must include at least one of the following:

  * Serum M-spike \>= 0.5 g/dL Patients with IgA, IgM or IgD myeloma in whom serum protein electrophoresis is deemed unreliable for routine M-protein quantitation may be considered eligible if total serum IgA, IgM or IgD level is elevated above normal range and parallels disease course
  * 24-hour urine M-spike \>= 200 mg
  * Involved serum free light chain (FLC) \>= 10 mg/L with an abnormal free light chain ratio
  * Bone marrow plasma cells \> 30%
  * Patients with extramedullary disease:

    * 1 lesion that has a single diameter of \>= 2 cm measured by computed tomography (CT) or magnetic resonance imaging (MRI) or the CT portion of the positron emission tomography (PET)/CT
    * Skin lesions can be used if the area is \>= 2cm in at least one diameter and measured with a ruler
* Willingness and ability to provide written informed consent
* Willing to return to Mayo Clinic Hospital in Arizona (MCA) for follow-up during the active monitoring phase of the study
* Willingness to provide mandatory blood, bone marrow biopsy, and aspirate specimens for correlative research
* Willing to undergo leukapheresis for blood component collection
* Absolute neutrophil count (ANC) \>= 1500/mm\^3 (within 14 days of study registration)
* Lymphocyte count \>= 500/mm\^3 (within 14 days of study registration)
* Hemoglobin \>= 8.0 g/dL (within 14 days of study registration)
* Platelet count \>= 30,000/mm\^3 (within 14 days of study registration)
* Total bilirubin =\< 2.0 mg/dL unless patient has documented Gilbert's syndrome (Subjects with Gilbert's Syndrome may be included if their total Bilirubin is =\< 3.0 x upper limit of normal (ULN) and direct bilirubin =\< 1.5 x ULN) (within 14 days of study registration)
* Alanine aminotransferase (ALT) and aspartate amino transferase (AST) =\< 3 x ULN (=\< 5 x ULN for patients with liver involvement of their cancer) (within 14 days of study registration)
* Prothrombin time, international normalized ratio (PT, INR) and activated partial thromboplastin time (aPTT) =\< 1.5 x ULN OR if patient is receiving anticoagulation therapy and INR or aPTT is within target range of therapy (for patients receiving anticoagulation, there should be no prior history of bleeding and no recent deep vein thrombosis \[DVT\]/pulmonary embolism \[PE\] within the last 6 months of enrollment)
* Calculated creatinine clearance \>= 30 ml/min using the Cockcroft-Gault formula (within 14 days of study registration)
* Baseline oxygen saturation \>= 90% on room air

Exclusion Criteria:

* Clinically unresolved central nervous system (CNS) metastases
* Prior treatment targeting MUC1
* Subjects with known plasma cell leukemia (PCL)
* Any of the following are excluded because this study involves an agent (CTX) that has known genotoxic, mutagenic and/or teratogenic effects:

  * Pregnant persons
  * Nursing persons
  * Persons of childbearing potential who are unwilling to employ adequate birth control measures
* History of myocardial infarction \>= 6 months prior to registration, and/or congestive heart failure requiring ongoing treatment such as medications and/or an implanted defibrillator to control life-threatening arrhythmias.
* Failure to recover to grade 1 or baseline from acute, reversible effects of prior therapy regardless of interval since last treatment.

EXCEPTION: Grade 1 peripheral (sensory) neuropathy that has been stable for at least 3 months since completion of prior treatment.

* Uncontrolled concurrent illness including, but not limited to:

  * Inability to clear an ongoing or active infection
  * Symptomatic congestive heart failure
  * Unstable angina pectoris
  * Uncontrolled psychiatric problems and/or difficult social situation
  * Dyspnea at rest due to complications of advanced malignancy or other disease that requires continuous oxygen therapy
  * Any other conditions that the protocol investigators deem could potentially limit compliance with study requirements
* Evidence of clinical immunocompromise and/or human immunodeficiency virus (HIV) positivity and currently receiving antiretroviral therapy
* Patients requiring chronic supraphysiologic daily doses of steroids (\>10 mg prednisone or prednisolone, \>= 4 mg Decadron or \>= 50 mg hydrocortisol daily)
* Patients receiving any other investigational agent which could be considered a treatment for the neoplasm
* Other active malignancy first documented =\< 4 years prior to registration. EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix. NOTE: If there is a history of other malignancy, the patient must not be receiving other treatment aimed at suppressing its recurrence.