Inclusion Criteria:

          -  Women or men with metastatic breast cancer with two or more bone metastases identified
             by technetium Tc-99m (99mTc) bone scintigraphy and/or computed tomography (CT), at
             least one of these bone lesions must not have been treated with prior radiation
             therapy

          -  A diagnosis of breast cancer must have been histologically or cytologically confirmed
             at any time point

          -  Patients with non-bone metastases (in addition to bone metastases) are permitted if:

               -  Five or less visceral metastasis (=< 4 cm in size) and asymptomatic (not
                  including lymph nodes)

               -  Enlarged lymph nodes =< 4 cm

          -  Patients with HER2 negative disease (HER2 negativity by immunohistochemistry [IHC] or
             fluorescent in situ hybridization [FISH] ratio according to the American Society of
             Clinical Oncology-College of American Pathologists guideline criteria) (Hammond et
             al., 2010; Wolff et al., 2013). Hormone-receptor positive (estrogen receptor
             [ER]-positive and/or progesterone receptor [PR]-positive) as well as triple-negative
             (ER-negative, PR-negative and no overexpression of HER2) breast cancer may be
             enrolled. Hormone receptor status will be determined at the local institution. ER and
             PR negativity will be defined as < 1% tumor staining by IHC

          -  Patient must be eligible to receive therapy with paclitaxel for the treatment of their
             breast cancer. Patients with hormone-receptor positive disease should have progressed
             on at least one prior line of hormone therapy and a CDK4/6 inhibitor in the metastatic
             setting to be eligible (except if patient had a contraindication or intolerable
             toxicity with the use of these agents). Previous radiation and chemotherapy for the
             treatment of metastatic breast cancer is allowed

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Absolute neutrophil count (ANC) >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (with the exception
             of < 3 mg/dL for patients with Gilbert's disease)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 3 x institutional ULN (=< 5 ULN for patients with liver metastasis)

          -  Creatinine =< 1.5 x institutional ULN OR glomerular filtration rate (GFR) >= 40
             mL/min/1.73 m^2

          -  Hemoglobin > 10 g/dL

          -  Human immunodeficiency virus (HIV)-infected patients on effective antiretroviral
             therapy with undetectable viral load within 6 months are eligible for this trial

          -  For patients with evidence of chronic hepatitis B virus (HBV) infection, 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 asymptomatic, treated brain metastases are permitted if there is no
             evidence of progression for at least 4 weeks after central nervous system
             (CNS)-directed treatment, as ascertained by clinical examination or brain imaging
             (magnetic resonance imaging [MRI] or CT scan) during the screening period

          -  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. History of other active
             malignancy requiring treatment within the last 3 years or bone marrow dysplasia such
             as myelodysplastic syndrome (MDS) is not allowed

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association functional classification. To be
             eligible for this study, patients should be class 2B or better

          -  Concomitant use of bisphosphonates or denosumab is required (except if medical
             contraindication such as hypocalcemia or concern for osteonecrosis of the jaw). If not
             already on bone modifying agents, patient must initiate such therapy within one month
             before start of study treatment

          -  The effects of radium-223 dichloride on the developing human fetus are unknown. For
             this reason and because alpha particle-emitting radiopharmaceutical agents as well as
             other therapeutic agents used in this trial are known to be teratogenic, women of
             child-bearing potential must agree to use adequate contraception (hormonal or barrier
             method of birth control; abstinence) prior to study entry, for the duration of study
             participation, and for at least 6 months after the last dose. Should a woman become
             pregnant or suspect she is pregnant while she or her partner is participating in this
             study, she should inform her treating physician immediately. Pre-menopausal subjects
             as well as subjects with ovarian radiation or concomitant treatment with an LH-RH
             agonist/antagonist must have a negative pregnancy test and agree to use an adequate
             method of contraception as recommended by their treating physicians. Subjects of
             child-bearing potential who are sexually active and their male partners must agree to
             utilize, during the treatment period and for 6 months after last dose of radium-223
             dichloride, 2 reliable and acceptable methods of contraception used simultaneously: a)
             barrier method such as a) condoms (male or female) with spermicidal agent or b)
             diaphragm or cervical cap with spermicide, combined with a highly effective
             non-hormonal birth control method such as an intra-uterine device. Men treated or
             enrolled on this protocol must also agree to use adequate contraception and not donate
             sperm prior to the study, for the duration of study participation, and 6 months after
             completion of Radium-223 dichloride

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible

          -  No prior paclitaxel in metastatic setting within 2 years prior to Radium-223
             dichloride start. No prior paclitaxel in adjuvant or neoadjuvant setting within 6
             months prior to Radium-223 dichloride start

        Exclusion Criteria:

          -  Patients with peripheral neuropathy > grade 1

          -  Patients who have not recovered from adverse events (AEs) due to prior anti-cancer
             therapy (i.e., have residual toxicities > grade 1) with the exception of alopecia

          -  Patients who have had chemotherapy or immunotherapy with checkpoint inhibitor within 4
             weeks prior to treatment. Patient who receives radiation therapy or hormone therapy
             within 2 weeks prior to treatment are excluded. For patients on trial therapy prior to
             study enrollment, washout period of 6 times the half-life of previously administered
             investigational agents prior to starting Radium-223 dichloride is required

          -  Prior therapy with radionuclides (e.g., strontium, samarium, rhenium, radium)

          -  Patients who are receiving any other investigational agents. Vaccination for severe
             acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is allowed as well as any
             therapy as required for the treatment of active coronavirus disease 2019 (COVID-19)
             infection

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to radium-223 dichloride or other agents used in study

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because radium-223 dichloride is an alpha
             particle-emitting radiopharmaceutical agent with the potential for teratogenic or
             abortifacient effects. Because there is an unknown but potential risk for AEs in
             nursing infants secondary to treatment of the mother with radium-223 dichloride,
             breastfeeding should be discontinued if the mother is treated with radium-223
             dichloride. These potential risks may also apply to other agents used in this study

          -  Imminent/established spinal cord compression, pathological fracture in weight bearing
             bones or bone lesion with soft tissue component unless treated as appropriate with
             radiation and/or surgery before starting on trial

          -  Prior hemibody external radiotherapy

          -  Patients must not have an active infection requiring systemic treatment

          -  Patients must not use immunosuppressive medication =< 7 days of registration, EXCEPT
             for the following:

               -  Intranasal, inhaled, topical steroids, or local steroid injection (e.g.,
                  intra-articular injection)

               -  Systemic corticosteroids at physiologic doses =< 10 mg/day of prednisone or
                  equivalent

               -  Steroids as premedication for hypersensitivity reactions (e.g., CT scan
                  premedication) is allowed

          -  Patients with Crohn's disease or ulcerative colitis

          -  Patients with a marked baseline prolongation of QT/corrected QT interval (QTc)
             interval (e.g., repeated demonstration of a QTc interval > 480 milliseconds [ms])
             (CTCAE grade 1) using Fridericia's QT correction formula

          -  Patients with a history of additional risk factors for Torsades de Pointes (TdP) (e.g.
             heart failure, hypokalemia, family history of long QT syndrome)

          -  The use of concomitant medications that prolong the QT/QTc interval

          -  Life expectancy < 6 months