  Primordial black holes (PBHs) produced in the early Universe have attracted
wide interest for their ability to constitute dark matter and explain the
compact binary coalescence. We propose a new mechanism of PBH production during
first-order phase transitions (PTs) and find that PBHs are naturally produced
during PTs model-independently. Because of the randomness of the quantum
tunneling, there always exists some probability that the vacuum decay is
postponed in a whole Hubble volume. Since the vacuum energy density remains
constant while radiation is quickly redshifted in the expanding Universe, the
postponed vacuum decay then results in overdense regions, which finally
collapse into PBHs as indicated by numerical simulations. Utilizing this result
one can obtain mutual predictions and constraints between PBHs and GWs from
PTs. The predicted mass function of PBHs is nearly monochromatic. We
investigate two typical cases and find that 1) PBHs from a PT constitute all
dark matter and GWs peak at $1$Hz, 2) PBHs from a PT can explain the
coalescence events observed by LIGO-Virgo collaboration, and meanwhile GWs can
explain the common-spectrum process detected by NANOGrav collaboration.
