Confirmatory trials increasingly test several treatments against a common control, and many balance prognostic factors through covariate-adaptive randomization (CAR). For time-to-event endpoints, they usually rely on the stratified log-rank test and the hazard ratio, but the former may lose power and the latter its clear interpretation when hazards are not proportional. The restricted mean survival time (RMST) provides an interpretable alternative, yet inference for multiple RMST comparisons under CAR remains largely unexplored. We first derive the joint asymptotic distribution of the standard Kaplan--Meier RMST estimators in multi-arm trials. Under Pocock--Simon minimization, their limiting covariance contains a design term that arises only when the randomization factors interact in their effect on survival. A stratum-standardized estimator removes this term and remains valid when censoring depends on the strata. We then develop test statistics that account for the shared control and the randomization design. Calibrated by a Gaussian multiplier, the resulting max-$T$ procedures control the family-wise error rate and give simultaneous confidence bands over restriction times. In simulations, the method controls the Type I error rate under various randomization procedures and outperforms the stratified log-rank test when the benefit wanes. An application based on the three-arm ICON6 trial illustrates its practicality.

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