quantum field theory - Is the sum of one-particle-irreducivle two-point diagrams always a real number?

On page 388 in section 11.6 of Peskin and Shroeder.
There appears an equation of the inverse propagator(the second functional derivative of the effective action) for a theory that contains several scalar fields:

$$K^2_{ij} =: \int d^4x e ^{ip\cdot (x-y)}\frac{\delta^2 \Gamma}{\delta \phi^i\delta \phi^j}(x,y)=0 \tag{11.105}$$ When diagonalizing
$$K^2_{ij} = P_{ik}P_{jl}\tilde{K}^2_{kl} = (P\tilde{K}P^t)_{ij} \, , \quad P \,\text{:an orthogonal matrix}$$ the property that $K^2_{ij}\,$: real is needed. After diagonalizing $$\tilde{K}^2_{ii}=p^2-m_{i0}^2-M_i^2(p^2) \quad\quad\text{i : no sum} \, ,$$ where $m_{i0}$ is the bare mass of the $i$th scalar field and $M_i^2(p^2)$ is the sum of one-particle-irreducivle two-point diagrams.
Is $M_i^2(p^2)$ always a real number?
That's simply because
$$\tilde{K}^2_{ii}=0 \quad \Leftrightarrow \quad m_i^2=m_{i0}^2-M_i^2(m_i^2) \quad ?$$ ,where $m_i$ is the physical mass.
Is that whole the story? Is there some other reason?
Thanks.