Let $M$ and $N$ be von Neumann algebras, and let $\rho:M\to N$ be a $*$-homomorphism. Say that $\rho$ is normal when it preserves suprema of bounded increasing nets in the positive cone: for every directed set $I$ and every bounded increasing net $(x_i)_{i\in I}$ in $M_+$ with supremum $x\in M_+$, the net $(\rho(x_i))_{i\in I}$ has supremum $\rho(x)$ in $N_+$. Then $\rho$ is normal if and only if the following projection-supremum condition holds: for every directed set $I$ and every increasing net of projections $(p_i)_{i\in I}$ in $M$ with supremum $p\in\mathcal P(M)$, the net of projections $(\rho(p_i))_{i\in I}$ has supremum $\rho(p)$ in $\mathcal P(N)$.