A Study of Multiple Immunotherapy-Based Treatment Combinations in Patients With Locally Advanced Unresectable or Metastatic Gastric or Gastroesophageal Junction Cancer (G/GEJ) or Esophageal Cancer (Morpheus-Gastric and Esophageal Cancer)

Open to people ages 18 years and up

Gastric Cancer Cohorts Inclusion Criteria:

• Age >/= 18 years;
• Eastern Cooperative Oncology Group (ECOG) Performance Status of 0 or 1;
• Life expectancy >/= 3 months, as determined by the investigator;

• Histologically or cytologically confirmed locally advanced unresectable or metastatic adenocarcinoma of gastric or gastroesophageal junction; (for the 1L

  Gastric Cancer Cohort: no prior systemic therapy for the locally advanced or metastatic disease; for the 2L Gastric Cancer Cohort: disease progression during or following a first-line platinum-containing or fluoropyrimidine-containing chemotherapy regimen);

• Availability of a representative tumor specimen that is suitable for determination of PD-L1 and TIGIT levels by IHC and/or additional biomarker status by means of retrospective central testing;
• Only for the 1L Gastric Cancer Cohort: human epidermal growth factor receptor 2 (HER2)-negative tumors;
• Measurable disease (at least one target lesion) according to Response Evaluation Criteria in Solid Tumors, Version 1.1 (RECIST v1.1);
• Adequate hematologic and end organ function based on laboratory results obtained within 14 days prior to initiation of study treatment;
• For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures as outlined for each specific treatment arm;
• For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm, as outlined for each specific treatment arm.

Esophageal Cancer Cohort Inclusion Criteria:

• Histologically or cytologically confirmed diagnosis of squamous cell carcinoma or adenocarcinoma of the esophagus in locally advanced or metastatic disease;
• No prior systemic treatment for esophageal cancer, with the following exception:

For patients treated with chemotherapy in the locally advanced setting: occurrence of metastasis after 6 months from the last dose of chemotherapy;

• For patients with adenocarcinoma: absence of HER2 expression;
• Life expectancy >/=3 months as determined by the investigator;
• Measurable disease per RECIST v1.1;
• Adequate hematologic and end-organ function;
• For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating eggs;
• For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm;
• ECOG Performance Status of 0, 1, or 2.

Exclusion criteria for the 2L Gastric Cancer Cohort:

• Urinary protein is > 1 + on dipstick and the required following 24-hour urine collection shows urinary protein > 2000 mg;
• Serious or non-healing wound, peptic ulcer, or bone fracture within 28 days prior to initiation of study treatment;
• History of gastrointestinal perforation and/or fistulae within 6 months prior to initiation of study treatment;
• Presence of a bowel obstruction, history or presence of inflammatory enteropathy, or extensive intestinal resection, Crohn disease, ulcerative colitis, or chronic diarrhea;
• Uncontrolled arterial hypertension >/= 150/ >/= 90 millimeter of mercury (mmHg) despite standard medical management;
• Chronic therapy with non-steroidal anti-inflammatory agents or other anti-platelet agents.

Gastric Cancer Exclusion Criteria:

• Uncontrolled hypercalcemia or symptomatic hypercalcemia requiring continued use of bisphosphonate therapy;
• Symptomatic, untreated, or actively progressing central nervous system (CNS) metastases;
• History of leptomeningeal disease;
• Active or history of autoimmune disease or immune deficiency;
• History of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on screening chest computed tomography (CT) scan;
• Positive test for human immunodeficiency virus (HIV) at screening;
• Active hepatitis B virus (HBV) or hepatitis C (HCV) infection;
• Severe infection within 4 weeks prior to initiation of study treatment;
• Significant cardiovascular disease;
• Significant bleeding disorder;
• Prior allogeneic stem cell or solid organ transplantation;
• Major surgical procedure, other than for diagnosis, within 4 weeks prior to initiation of study treatment, or anticipation of need for a major surgical procedure during the study;
• Treatment with anticoagulation with warfarin, low-molecular-weight heparin, or similar agents for therapeutic purposes;
• History of malignancy other than gastric or gastroesophageal junction carcinoma within 2 years prior to screening, with the exception of those with a negligible risk of metastasis or death;
• Known allergy or hypersensitivity to any of the study drugs or their excipients.

Esophageal Cancer Cohort Exclusion Criteria:

• High risk for developing esophageal fistula by clinical assessment or imaging;
• Symptomatic, untreated, or actively progressing central nervous system (CNS) Metastases;
• Positive EBV viral capsid antigen IgM test at screening;
• History of leptomeningeal disease;
• Active or history of autoimmune disease or immune deficiency;
• History of idiopathic pulmonary fibrosis, organizing pneumonia, drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on screening chest computed tomography (CT) scan;
• Active tuberculosis;
• Significant cardiovascular disease within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina;
• History of malignancy other than esophageal cancer within 2 years prior to screening, with the exception of malignancies with a negligible risk of metastasis or death;
• Pregnancy or breastfeeding, or intention of becoming pregnant during study treatment or within 5 months after the final dose of atezolizumab or within 90 days after the final dose of tiragolumab.