Scalar field theory with two scalars Consider the following scalar field theory with a kinetic term as follows
$$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_1\partial^{\mu}\phi_1-\frac{1}{2}\partial_{\mu}\phi_2\partial^{\mu}\phi_2 + V(\phi_1, \phi_2).$$
Notice the $-$ sign used in the second kinetic term. Is this a valid QFT? If we were to redefine the field variables as $\phi_{\pm} = \phi_1\pm\phi_2$, then
$$ \mathcal{L} = \frac{1}{2}\partial_{\mu}\phi_+\partial^{\mu}\phi_- + V(\phi_+, \phi_-).$$
We see that this almost looks like a complex scalar field theory which it isn't. And in this field-redefinition one of the field seems to be non-dynamical (if you used integration by parts and shifted the derivatives around). Are such QFTs allowed? Are there examples of such fields in particle physics?
 A: The kinetic term in OP's model is not positive definite/contains ghosts. The corresponding Hamiltonian $H$ is not bounded from below.
In quantum theory we usually require that the Hamiltonian $H$ is bounded from below and that the system has a ground state. This is intimately related to unitarity.
A: Particles with negative kinetic energy are usually known in the literature as ghosts. While in some cases they can be dealt with, they do have some spooky properties.
As noted in Qmechanic's answer, this theory has a Hamiltonian that is unbounded from both above and below. A problematic thing about this is that it allows for the vacuum of the theory to decay: if you have modes with negative energy, then the vacuum can decay to a state $|\mathbf{p}_{\text{ord}} \mathbf{q}_{\text{gh}}\rangle,$ composed of an ordinary particle and a ghost. Since the ghost still has negative energy, the total energy is still zero. Notice also that there is a ridiculously large amount of ways this decay can happen: the new particles can have any energy whatsoever (as long as they cancel out) and so on. This implies that the decay rate will often diverge, meaning the vacuum is unstable to the point it decays immediately. Of course, the same argument applies to any other state. Hence, in this sense, the theory becomes unstable.
There are some contexts in which these objects appear in quantum field theory, although they are usually undesirable and one often wants to bust the ghosts. A notable example is that of the Ostrogradsky instability, in which the presence of higher-derivative terms on the Lagrangian lead to the occurrence of ghosts in the theory. A recent review on this topic which uses a multitude of examples from Classical Mechanics and discusses applications to theories of modified gravity is arXiv: 2007.01063 [hep-th]. Another interesting reference is arXiv: hep-th/0107088, which is a famous paper by Hawking and Hertog discussing the occurrence of ghosts in quantum gravity and how to "live with them".
A: The kinetic part very much resembles a relativistic scalar field theory on a Schwinger-Keldysh contour. In this context, the transformation you have performed is often referred to as the "Keldysh rotation", while $+$ and $-$ fields are usually called classical and quantum components, respectively. There are certain caveats, however: for instance, one should then also have (at least infinitesimal) $\sim \phi_{-} \phi_{-}$ term, etc. For details, see any decent literature on non-equilibrium QFT (Kamenev, Calzetta-Hu, etc.).
