  The classical $J_1$-$J_2$ Ising model on the square lattice is a minimal
model of frustrated magnetism whose phase boundaries have remained under
scrutiny for decades. Signs of first-order phase transitions have appeared in
some studies, but strong evidence remains lacking. The current consensus, based
upon the numerical data and theoretical arguments in [S. Jin et al., Phys. Rev.
Lett. \textbf{108}, 045702 (2012)], is that first-order phase transitions are
ruled out in the region $g = J_2/|J_1|\gtrsim 0.67$. We point out a loophole in
the basis for this consensus, and we find strong evidence that the phase
boundary is instead weak first-order at $0.67\lesssim g<\infty$ such that it
asymptotically becomes second-order when $g\rightarrow\infty$. We also find
strong evidence that the phase boundary is first-order in the region
$0.5<g\lesssim0.67$. We establish these results with adiabatic evolution of
matrix product states directly in the thermodynamic limit, and with the theory
of finite entanglement scaling. We also find suggestive evidence that when
$g\rightarrow0.5^+$, the phase boundary becomes of an anomalous first-order
type wherein the correlation length is very large in one of the coexisting
phases but very small in the other.
