  Fast radio bursts (FRBs) represent an exciting frontier in the study of
gravitational lensing, due to their brightness, extragalactic nature, and the
compact, coherent characteristics of their emission. In a companion work
[Kader, Leung+2022], we use a novel interferometric method to search for
gravitationally lensed FRBs in the time domain using bursts detected by
CHIME/FRB. There, we dechannelize and autocorrelate electric field data at a
time resolution of 1.25 ns. This enables a search for FRBs whose emission is
coherently deflected by gravitational lensing around a foreground compact
object such as a primordial black hole (PBH). Here, we use our non-detection of
lensed FRBs to place novel constraints on the PBH abundance outside the Local
Group. We use a novel two-screen model to take into account decoherence from
scattering screens in our constraints. Our constraints are subject to a single
astrophysical model parameter -- the effective distance between an FRB source
and the scattering screen, for which we adopt a fiducial distance of 1 parsec.
We find that coherent FRB lensing is a sensitive probe of sub-solar mass
compact objects. Having observed no lenses in $172$ bursts from $114$
independent sightlines through the cosmic web, we constrain the fraction of
dark matter made of compact objects, such as PBHs, to be $f \lesssim 0.8$, if
their masses are $\sim 10^{-3} M_{\odot}$.
