= Solution
For one bosonic mode, define the <unnormalized bosonic coherent state> $|z\rangle=e^{z a^\dagger}|0\rangle$. The <commutator> $[a,a^\dagger]=1$ gives $a|z\rangle=z|z\rangle$ and $\langle z'|z\rangle=e^{\bar z'z}$. The normalized state is $e^{-|z|^2/2}|z\rangle$; it is not orthogonal to a different <coherent state>. The <coherent-state resolution of identity> follows from Gaussian moments,
$$
\int\frac{d^2z}{\pi}e^{-|z|^2}z^n\bar z^m=n!\delta_{nm},\qquad \boxed{\mathbf1=\int\frac{d^2z}{\pi}e^{-|z|^2}|z\rangle\langle z|.}
$$
For many modes take products of these states and measures. The complex amplitudes have the natural scale of square root occupation; <coherent states> turn normal-ordered <matrix elements> into classical symbols without replacing the quantum <trace> by a single classical configuration.
For <fermions>, $\{c_i,c_j^\dagger\}=\delta_{ij}$ and a nilpotent <annihilation operator> cannot have an arbitrary nonzero ordinary complex <eigenvalue>. Introduce independent <Grassmann variables> $\eta_i,\bar\eta_i$, anticommuting among themselves and with the fermionic operators, and define $|\eta\rangle=\exp(-\sum_i\eta_i c_i^\dagger)|0\rangle$. Then $c_i|\eta\rangle=\eta_i|\eta\rangle$ and $\langle\bar\eta'|\eta\rangle=e^{\sum_i\bar\eta_i'\eta_i}$. With an ordered <Berezin integral> measure satisfying $\int d\bar\eta\,d\eta\,\bar\eta\eta=-1$ for one pair,
$$
\mathbf1=\int D(\bar\eta,\eta)e^{-\bar\eta\eta}|\eta\rangle\langle\bar\eta|,\qquad \operatorname{Tr}O=\int D(\bar\eta,\eta)e^{-\bar\eta\eta}\langle-\bar\eta|O|\eta\rangle
$$
for an even operator $O$; this is the <fermionic coherent-state trace>. In one mode the <trace> integrand is $e^{-\bar\eta\eta}(O_{00}-\bar\eta\eta O_{11})$, whose integral is $O_{00}+O_{11}$. Without the sign reversal it would be the <supertrace>, not the thermal <trace>. These formal Grassmann-valued <coherent states> do not represent ordinary complex-valued <fermion> amplitudes.
To construct the many-body thermal <path integral>, write $\mathcal Z=\operatorname{Tr}e^{-\beta K}$ with $K=H-\mu\hat N$ and divide the imaginary-time interval $\mathcal T=\beta\hbar$ into slices of width $\delta\tau$. Insert the appropriate <coherent-state resolution of identity> between factors. For a normal-ordered <Hamiltonian>, the adjacent-slice kernel is
$$
\langle\bar\psi_{j+1}|e^{-\delta\tau K/\hbar}|\psi_j\rangle=\exp\left[\bar\psi_{j+1}\psi_j-\frac{\delta\tau}{\hbar}K_N(\bar\psi_{j+1},\psi_j)+O(\delta\tau^2)\right].
$$
<coherent-state time slicing> makes the overlap and <Gaussian measure> combine into the first-order time derivative. The <grand-canonical coherent-state action> in the regulated <continuum limit> is
$$
\boxed{\mathcal Z=\int\mathcal D(\bar\psi,\psi)\,e^{-S_E/\hbar},\qquad S_E=\int_0^{\beta\hbar}d\tau\left[\hbar\sum_i\bar\psi_i\partial_\tau\psi_i+H_N(\bar\psi,\psi)-\mu\sum_i\bar\psi_i\psi_i\right].}
$$
For <bosons> the integration variables are complex, with the conjugate labels on the original integration contour; for <fermions> the barred and unbarred <Grassmann variables> are independent. The continuum notation retains the time-sliced ordering prescription. For example, $n(n-1)=a^{\dagger2}a^2$ has normal symbol $\bar z^2z^2$, whereas $n^2$ has that symbol plus $\bar zz$. Dropping that distinction changes a theory. The <coherent-state thermal boundary conditions> follow because the bosonic <trace> closes paths periodically and the fermionic sign in the <trace> closes them antiperiodically:
$$
\psi(\beta\hbar)=\psi(0)\quad\text{(bosons)},\qquad \psi(\beta\hbar)=-\psi(0)\quad\text{(fermions)}.
$$
The barred fields obey the same respective boundary conditions. Hence <Matsubara frequencies> are $2\pi n/(\beta\hbar)$ and $(2n+1)\pi/(\beta\hbar)$. The derivative term encodes the canonical structure; omitting it would remove the quantum dynamics.<Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§§3.1–3.2, printed pp.51–61: coherent-state completeness, time slicing and fermionic trace boundary conditions]{ref}
As a check, a free diagonal mode with $K=\xi a^\dagger a$ gives $Z_B=(1-e^{-\beta\xi})^{-1}$ for $\xi>0$, whereas $K=\xi c^\dagger c$ gives $Z_F=1+e^{-\beta\xi}$. Gaussian integration produces an inverse <determinant> for <bosons> and a <determinant> for independent complex <fermions>, with their respective temporal boundary conditions. Differentiating $\log Z$ recovers Bose-Einstein and Fermi-Dirac occupations. Interactions become higher powers in the normal-ordered symbols, and sources generate time-ordered correlations. \b[The coherent-state construction preserves statistics, ordering and boundary conditions together; a formal continuum expression without its measure and temporal prescription is incomplete.]
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