So this is really the twist in the construction of QFT that is still sometimes misunderstood. Historically, people like Dirac noticed that solutions to relativistic field equations such as the spinor equation for $\psi(x)$ admitted apparent negative mass-energy densities. However, once you go into QFT, either from a "second quantization" or a Fockian bottom-up construction, you realize that these were never negative energy densities, it is just that $\psi(x)$ represented a "mirrored mix" of creation operators of one type of particles and annihilation operators for their "mirror buddies". This means that operators such as $\psi(x)$ acting on a state would create a particle at $x$, and destroy its mirror buddy, if it was at $x$.
Long story short, "mirror buddies", also known as "antiparticles" are absolutely and definitely particles with positive energy, the same as "ordinary particles", and they cannot spring into existence without a positive source of energy. (Let's not talk too much about "virtual particles" which allow quantum weirdness influence interactions just because of the "possibility of particles/antiparticles being there, if enough energy was present".)
More specifically, the construction of QFT requires postulating a vacuum, and its definition is that you cannot just freely pump more positive-energy particles/antiparticles from it, since negative energies are not allowed. This assumption might not be obvious, but it is actually simply the only choice that allows you to obtain any useful predictions from the theory. And one last thing - there is no meaningful way in which a positive energy particle travelling forward in time can be distinguished from a negative energy particle travelling backwards in time. So you could say all the matter falling into a black hole is negative-energy matter traveling backwards in time. But you will never see a positive energy particle travelling backwards in time out of a black hole.
The reason why antiparticles can sometimes be referred to as "time-reversed particles" is a bit subtle and has to do with Lorentz symmetry, particularly CPT. It is just that you are allowed to build theories which have time-irreversible processes while staying Lorentz-invariant as long as you have those "mirror buddies" in the game as well. The point is that the theory does not need to allow the identical time-reversed processes with the original particles, it just needs to allow identical time-reversed processes with their mirror buddies (and possibly also reversed parity).
I kind of understand where you are coming from with this question - it really goes back to the popular understanding of the creation of Hawking radiation by virtual particle-antiparticle pairs. But in this explanation it is not important that the particle going "in" is an antiparticle, while the one going "out" is a particle. You can swap those two and, indeed, Hawking radiation is made of both particles and antiparticles. What allows Hawking radiation is the "quantum weirdness" that allows the quantum principles to virtually "borrow" energy-momentum and transfer it over space-like intervals if it is energetically favorable. In the language of virtual particles, it is a particle-antiparticle pair either of which is executing the transport in either direction.
Without going any deeper into the details, I just want to point out that the only thing that allows the black hole to "leak" particles is the breaking of time reversal by the collapse. A perfectly time-reversible computation actually yields no radiation (which was actually computed by Unruh some time before Hawking). So really Hawking radiation is created despite CPT symmetry.
