Inclusion Criteria:

          -  PHASE 1: Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky
             >= 70%)

          -  PHASE 2: ECOG performance status =< 2 (Karnofsky >= 60%)

          -  Unless a patient has had orchiectomy by surgery, the patient is expected to be on
             antiandrogen therapy (ADT) for "medical castration". ADT needs to be maintained
             throughout the study. Testosterone level should be checked, and kept consistently
             lower than 50 ng/dL, similar to that obtained with bilateral orchiectomy

          -  Progressive castration-resistant prostate cancer with two or more skeletal metastases
             identified by 99mTC bone scintigraphy. One or more lymph node metastases allowed, but
             not mandatory. Lymph node metastases in each individually must measure less than 3 cm
             in the longest dimension. Visible visceral organ metastases are not allowed. A
             diagnosis of prostate cancer must have been histologically confirmed at any time point

          -  Baseline prostatic specific antigen (PSA) level of 1 ng/mL or higher with evidence of
             progressively increasing PSA values (two consecutive increases over the previous
             reference value)

          -  Progression after at least one of the following: abiraterone, enzalutamide,
             apalutamide, darolutamide, or taxane chemotherapy (docetaxel, cabazitaxel). There is
             no maximum number of prior therapies

          -  Life expectancy >= 6 months

          -  Albumin > 2.5 mg/dL

          -  Hemoglobin > 9 mg/dL

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 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

          -  Creatinine =< 1.5 x institutional ULN OR

          -  Glomerular filtration rate (GFR) >= 40 mL/min/1.73 m^2

          -  Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
             therapy (with no medications prohibited by this protocol [e.g. drug-drug
             interactions]) with undetectable viral load within 6 months 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 (with no medications prohibited by
             this protocol [e.g. drug-drug interactions]), 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 (with no medications prohibited
             by this protocol [e.g. drug-drug interactions])

          -  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

          -  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 trial, patients should be class 2B or better

          -  Concomitant use of physiologic corticosteroids is allowed

          -  Concomitant use of bisphosphonates is allowed (use of bone health agents is mandatory
             - either denosumab [preferred] or bisphosphonates)

          -  The effects of radium-223 dichloride, M3814, and avelumab on the developing human
             fetus are unknown. Men treated or enrolled on this protocol must also agree to use
             adequate contraception prior to the study, for the duration of study participation,
             and 6 months after completion of Radium-223 dichloride, M3814, and avelumab
             administration

          -  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

          -  Patients must be able to swallow orally administered medication

          -  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 and brain imaging
             (magnetic resonance imaging [MRI] or computed tomography [CT] scan) during the
             screening period

        Exclusion Criteria:

          -  Active autoimmune conditions or patients on chronic immunosuppression due to
             underlying autoimmune condition

          -  Patients who have had chemotherapy or radiotherapy within 4 weeks prior to entering
             the study

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

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

          -  Patients who are receiving any other investigational agents

          -  Patients who have had previous hemibody external radiation

          -  Patients who have had systemic radiotherapy with radioisotopes

          -  Patients who have 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

          -  Patients who have a history of allergic reactions attributed to compounds of similar
             chemical or biologic composition to radium-223 dichloride, M3814, or avelumab

          -  Patients unable to discontinue medications or substances that are potent inhibitors,
             inducers or sensitive substrates of CYP3A4/5 or CYP2C19 prior to study treatment are
             ineligible. Medications or substances that are strong inhibitors of CYP3A4/5 or
             CYP2C19 must be discontinued at least 1 week prior to first M3814 dose. Strong
             inducers of CYP3A4/5 or CYP2C19 must be stopped at least 3 weeks prior to the first
             dose. Drugs mainly metabolized by CYP3A with a narrow therapeutic index as judged by
             the Investigator must stop at least 1 day prior to first M3814 dose. Because the lists
             of these agents are constantly changing, it is important to regularly consult a
             frequently- updated medical reference. As part of the enrollment/informed consent
             procedures, the patient will be counseled on the risk of interactions with other
             agents, and what to do if new medications need to be prescribed or if the patient is
             considering a new over-the- counter medicine or herbal product. The primary
             elimination mechanism of avelumab is proteolytic degradation, thus there are no
             contraindicated medications with respect to avelumab. Radium-223 dichloride should not
             be given concurrently with abiraterone plus prednisone/prednisolone

          -  Patients with uncontrolled intercurrent illness

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

          -  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)

          -  Patients who cannot discontinue concomitant H2 blockers or proton-pump inhibitors
             (PPIs). Patients may confer with the study doctor to determine if such medications can
             be discontinued. These must be discontinued >= 5 days prior to study treatment.
             Patients do not need to discontinue calcium carbonate

          -  Patients receiving sorivudine or any chemically related analogues (such as brivudine)
             are excluded

          -  Patients with a known history or present osteonecrosis of the jaw