Inclusion Criteria:

• Histologically or cytologically confirmed diagnosis of locally-advanced unresectable or metastatic HER2+ gastric or gastroesophageal junction adenocarcinoma.
• HER2+ disease documented since progression of the most recent line of systemic therapy, as follows:
  • Phase 2 paclitaxel dose optimization stage:
  • HER2 amplification in a blood-based NGS assay performed at a central laboratory; or
  • HER2 overexpression/ amplification by IHC and ISH (IHC3+ or IHC2+/ISH+) assay of a tumor tissue sample, processed in a Clinical Laboratory Improvement Amendments (CLIA)- or International Organization for Standardization (ISO)‑accredited laboratory;
  • Phase 2 dose expansion stage:
  • Cohort 2A: HER2 amplification in a blood-based NGS assay performed at a central laboratory;
  • Cohort 2B: No HER2 amplification by blood-based NGS assay, but HER2 overexpression/amplification by IHC and ISH (IHC3+ or IHC2+/ISH+) assay of a tumor tissue sample, processed in a CLIA- or ISO‑accredited laboratory;
  • Phase 3: HER2 amplification in a blood-based NGS assay performed at a central laboratory.
• Can supply archival tumor tissue for central review; if an archival sample is not available, the subject may be eligible, following approval by the medical monitor.
• History of prior treatment with a HER2-directed antibody.
• Progressive disease during or after first-line therapy for locally-advanced unresectable or metastatic GEC 6.
• Phase 2: Measurable disease according to RECIST version 1.1.
• Phase 3: Measurable or non-measurable disease according to RECIST version 1.1 7.
• Age ≥ 18 years, or considered an adult by local regulations, at time of consent.
• Eastern Cooperative Oncology Group (ECOG) performance status score of 0 or 1.
• Life expectancy of at least 3 months, in the opinion of the investigator.
• Adequate hepatic function as defined by the following:
  • Total bilirubin ≤ 1.5 × upper limit of normal (ULN), except for subjects with known Gilbert’s disease, who may enroll if the conjugated bilirubin is ≤ 1.5 × ULN;
  • Transaminases (aspartate aminotransferase [AST] and alanine aminotransferase [ALT]) ≤ 2.5 × ULN (≤ 5 × ULN if liver metastases are present).
• Adequate baseline hematologic parameters as defined by:
  • Absolute neutrophil count ≥ 1.5 × 10^9 /L;
  • Platelet count ≥ 100 × 10^9 /L; subjects with stable platelet count from 75-100 × 10^9 /L may be included with approval from the Medical Monitor;
  • Hemoglobin ≥ 9 g/dL; subjects with hemoglobin ≥ 8-9 g/dL may be included with approval from the Medical Monitor;
  • In subjects transfused before study entry, transfusion must be ≥ 14 days prior to start of therapy to establish adequate hematologic parameters independent from transfusion support.
• Estimated glomerular filtration rate (GFR) ≥ 50 mL/min/1.73 m² using the Modification of Diet in Renal Disease (MDRD) study equation as applicable.
• International normalized ratio (INR) ≤ 1.5, and prothrombin time (PT) and partial thromboplastin time (PTT)/activated partial thromboplastin time (aPTT) ≤ 1.5 × ULN if not receiving anticoagulation therapy. Subjects on full-dose anticoagulation must be on a stable dose of oral anticoagulant or low molecular weight heparin. If on warfarin, the subject must have an INR of ≤ 3 and have no active bleeding (within ≤ 14 days prior to enrollment or randomization, excluding trace hematuria) or pathological condition that carries a high risk of bleeding (i.e., tumor involving major vessels or known varices). Therapy for potential drug interactions with tucatinib and oral anticoagulants.
• Left ventricular ejection fraction (LVEF) ≥ 50% as assessed by echocardiogram or multi-gated acquisition (MUGA) scan documented within 4 weeks prior to first dose of study treatment .
• Urinary protein of ≤ 1+ on dipstick or routine urinalysis. If dipstick or routine analysis indicates proteinuria ≥ 2+, then either a 24-hour urine must be collected and must demonstrate < 1000 mg of protein in 24 hours or the urine protein/creatinine (UPC) ratio must be <1 to allow participation in study.
• For subjects of childbearing potential, the following stipulations apply:
  • Must have a negative serum or urine pregnancy test (minimum sensitivity of 25 mIU/mL or equivalent units of beta human chorionic gonadotropin [β-hCG]) result within 7 days prior to the first dose of study treatment. A subject with a false positive result and documented verification that the subject is not pregnant is eligible for participation;
  • Must agree not to try to become pregnant during the study and for at least 7 months after the final dose of any study drug;
  • Must agree not to breastfeed or donate ova, starting at time of informed consent and continuing through 7 months after the final dose of any study drug;
  • If sexually active in a way that could lead to pregnancy, must consistently use 2 highly effective methods of birth control, starting at the time of informed consent and continuing throughout the study and for at least 7 months after the final dose of any study drug.
• For subjects who can father children, the following stipulations apply:
  • Must agree not to donate sperm starting at time of informed consent and continuing throughout the study period and for at least 7 months after the final dose of any study drug;
  • If sexually active with a person of childbearing potential in a way that could lead to pregnancy, must consistently use 2 highly effective methods of birth control, starting at time of informed consent and continuing throughout the study and for at least 7 months after the final dose of any study drug;
  • If sexually active with a person who is pregnant or breastfeeding, must consistently use one of 2 contraception options, starting at time of informed consent and continuing throughout the study and for at least 7 months after the final dose of any study drug.
• The subject must provide written informed consent.
• Subject must be willing and able to comply with study procedures, laboratory tests, and other requirements of the study

Exclusion Criteria:

• Subjects with squamous cell or undifferentiated GEC.
• Having received more than 1 line of prior systemic therapy for locally-advanced unresectable or metastatic disease.
• Having received taxanes ≤ 12 months prior to enrollment, prior treatment with ramucirumab, or prior treatment with tucatinib, lapatinib, neratinib, afatinib, or any other investigational anti-HER2 and/or anti-EGFR tyrosine kinase inhibitor, or with trastuzumab emtansine (T-DM1), trastuzumab deruxtecan (T-DXd), or any other HER2-directed antibody-drug conjugate.
• History of exposure to the following cumulative doses of anthracyclines:
  • Doxorubicin >360 mg/m²;
  • Epirubicin >720 mg/m²;
  • Mitoxantrone >120 mg/m²;
  • Idarubicin >90 mg/m²;
  • Liposomal doxorubicin (e.g., Doxil, Caelyx, Myocet) > 550 mg/m².
  • History of allergic reactions to paclitaxel, trastuzumab, ramucirumab, or compounds chemically or biologically similar to tucatinib, except for Grade 1 or 2 infusion-related reactions to trastuzumab or ramucirumab that were successfully managed, or known allergy to any of the excipients in the study drugs or placebos.
• Phase 2 paclitaxel dose optimization stage only: history of prior partial or total gastrectomy
• Treatment with any systemic anti-cancer therapy (including hormonal and biologic therapy), radiation, or an experimental agent, or participation in another interventional clinical trial ≤ 3 weeks prior to first dose of study treatment.
• Major surgery within 28 days prior to enrollment or randomization, central venous access device placement within 7 days prior to enrollment or randomization, or planned major surgery following initiation of study treatment.
• Any toxicity related to prior cancer therapies that has not resolved to ≤ Grade 1, with the following exceptions:
  • Anemia;
  • Alopecia;
  • Congestive heart failure (CHF), which must have been ≤ Grade 1 in severity at the time of occurrence, and must have resolved completely.
• Clinically significant cardiopulmonary disease such as:
  • Ventricular arrhythmia requiring therapy;
  • Symptomatic hypertension or uncontrolled asymptomatic hypertension ≥ 150/≥90 mmHg despite standard medical management, as determined by the investigator;
  • Any symptomatic history of CHF, left ventricular systolic dysfunction or decrease in ejection fraction;
  • Severe dyspnea at rest (Common Terminology Criteria for Adverse Events [CTCAE] Grade 3 or above) due to complications of advanced malignancy or hypoxia requiring supplementary oxygen therapy, except when therapy is needed for obstructive sleep apnea.
• Known to be positive for hepatitis B by surface antigen expression. Known to be positive for hepatitis C infection (positive by polymerase chain reaction). Subjects who have been treated for hepatitis C infection are permitted if they have documented sustained virologic response of 12 weeks.
• Presence of known chronic liver disease.
• Phase 2: Known to be positive for human immunodeficiency virus (HIV).
• Phase 3: Subjects known to be positive for HIV are excluded if they meet any of the following criteria:
  • CD4+ T-cell count of < 350 cells/uL;
  • Detectable HIV viral load;
  • History of an opportunistic infection within the past 12 months;
  • On stable antiretroviral therapy for < 4 weeks.
• Subjects who are pregnant, breastfeeding, or planning to become pregnant from time of informed consent until 7 months following the last dose of study drug.
• Unable to swallow pills.
• Have used a strong cytochrome P450 (CYP)2C8 inhibitor within 5 half-lives of the inhibitor, or have used a strong CYP2C8 or CYP3A4 inducer within 5 days prior to first dose of study treatment.
• Other medical, social, or psychosocial factors that, in the opinion of the investigator, could impact safety or compliance with study procedures