Evaluate the Safety, Immunologic, and Virologic Responses of Donor Derived (DD) HIV-Specific T-cells (HST) in HIV-infected Individuals Following Allogeneic Bone Marrow Transplantation (alloRESIST)

Inclusion Criteria: Participant Inclusion Criteria at Screening: * Age ≥18 years. * Confirmation of HIV-1 infection. Any licensed ELISA test kit which is confirmed by Western blot or Multispot HIV-1/HIV-2 assay prior to screening. HIV culture, HIV antigen, plasma HIV RNA, or a second antibody test by a method other than ELISA is acceptable as an alternative confirmatory test. * On effective antiretroviral therapy. * Ability and willingness of participant to continue and be compliant with ART throughout the study. * Hematologic malignancy that qualifies for standard of care alloBMT according to JHU criteria. * Potential participant must have adequate organ function for standard of care alloBMT according to JHU criteria. * No active HCV infection. (If seropositive, participant must have no measureable HCV RNA within 30 days of enrollment). * No active HBV infection (If seropositive, participant must have no measureable HBV DNA or HBsAg+ within 30 days of enrollment). * Ability and willingness of participant to give written informed consent. * Ability and willingness to communicate effectively with study personnel; considered reliable, willing, and cooperative in terms of compliance with the Protocol requirements. * Ability and willingness to provide adequate locator information and contact information for at least 2 adults who can reach the participant within 24 hours Participant Inclusion Criteria for DD HST-NEETs Infusion: * Karnofsky score of ≥ 70. * ANC ≥ 250/µL. * Bilirubin ≤ 2x upper limit normal or direct bilirubin normal. * AST ≤ 3x upper limit normal. * Serum creatinine ≤ 2x upper limit normal. * Hgb ≥ 7.0 g/dL. * Pulse oximetry of \> 90% on room air. * Negative pregnancy test in female participants if applicable (childbearing potential). * Written informed consent signed by participant or guardian. * Steroids less or equal to 0.5 mg/kg/day prednisone. Donor Inclusion Criteria for Procurement for DD HST-NEETS Manufacturing: * Donors for allogeneic (i.e. HLA matched or mismatched related or unrelated) hematopoietic cell transplants who have fulfilled eligibility for and consented to stem cell donation as per JHU standard operating procedures. * Donor must be in good health based on institutional guidelines. * Female donors of childbearing age must have a negative pregnancy test and must not be lactating. * It is understood that medical clearance from the donor will be sought within the timeline per the National Marrow Donor Program (NMDP) rules. * The hematopoietic cell donor will have already been selected by the JHU BMT Donor Selection Committee. * Donor or parent/guardian capable of providing informed consent Exclusion Criteria: Participant Exclusion Criteria DD HST-NEETs Infusion: * Participants receiving ATG, or Campath or other immunosuppressive T-cell monoclonal antibodies within 28 days. * Participants with uncontrolled infections. For bacterial infections, participants must be receiving definitive therapy and have no signs of progressing infection for 72 hours prior to enrollment. For fungal infections participants must be receiving definitive systemic anti-fungal therapy and have no signs of progressing infection for 1 week prior to enrollment. Progressing infection is defined as hemodynamic instability attributable to sepsis or new symptoms, worsening physical signs or radiographic findings attributable to infection. * Participants who have received donor lymphocyte infusion (DLI) within 28 days. * Active and uncontrolled relapse of malignancy. * Participants with active acute GVHD grades II-IV * Participants with bronchiolitis obliterans syndrome or serositis * Any licensed or experimental non-HIV vaccination (e.g., hepatitis B, pneumococcal polysaccharide) within 28 days prior to study entry. * Inability to comply with study requirements, which could impact study integrity and/or safety. Donor Exclusion Criteria for Procurement for DD HST-NEETs Manufacturing: • Donor exclusion criteria will be followed as per institution standard operating procedures (SOPs).