I have heard that in QFT, the vacuum has zero four-momentum: $$P^\mu |\Omega \rangle = 0.$$ However, I also know that the vacuum has vacuum energy, i.e. $$ \langle \Omega | H | \Omega \rangle = E_0 \neq 0.$$ In textbooks, we patch up the latter equation by subtracting an infinite constant, and are told not to worry about it. In any case, the vacuum has to have zero four-momentum, as a fundamental axiom.

I am confused since a lot of answers on this site talk about vacuum energy as a real thing that has a definite value. If it's real, then why can we subtract it off to get $P^\mu |\Omega \rangle = 0$? If it's not real, why do we talk about it as if it is?