We demonstrate single-shot temporal-based interferometric phase measurements for bursts of 35 pulses of 1ns duration in an Yb-doped fiber amplifier chain seeded by a 1063nm single-frequency diode laser and operating in a highly nonlinear and saturated regime. We determined a maximum intra-pulse phase distortion (B-integral) of 21.2 rad by mixing the burst with a reference. The difference in the B-integral between the first and last pulse in the burst was large, 12.8 rad, but can be compensated for. Moreover, burst-to-burst amplitude fluctuations (standard deviation, std) were below 6%, where phase fluctuations were ~0.6 rad (std) or less. Even if these fluctuations cannot be compensated for, they still allow for efficient coherent combination, which opens up for transform-limited or narrow-line high-energy divided-pulse amplification with single-frequency seeding.