The following fix seems to work:
using MatrixMarket: mmread
using Random: seed!
using IncompleteLU: ilu
using LinearAlgebra: I
A = mmread("transient.mtx");
dA = 10^-6
M = ilu(A+dA*I);
seed!(1);
b = rand(A.n);
sum(isnan.(M \ b))
Then I get zero NaN, and I can successfully use M in GMRES.