= Solution
The naive equal-time <canonical commutation relations> would be
$$
[B^\mu(t,\mathbf x),\pi_\nu(t,\mathbf y)]=i\delta^\mu{}_{\nu}\delta^{(3)}(\mathbf x-\mathbf y),\qquad[B^{*\mu}(t,\mathbf x),\pi_\nu^*(t,\mathbf y)]=i\delta^\mu{}_{\nu}\delta^{(3)}(\mathbf x-\mathbf y),
$$
with all coordinate-coordinate, momentum-momentum and mixed conjugate-sector commutators zero before enforcing <constraints>. But the $\mu=\nu=0$ relation contradicts $\pi_0=0$ as an operator identity: its left side would vanish and its right side would not.
The primary zero-momentum <constraints> and the secondary massive <Gauss laws> form <second-class constraints>; their <Poisson bracket> matrix contains nonzero terms proportional to $M^2$. They must be eliminated, or treated using a <Dirac bracket>, before quantization. In a reduced description the three complex spatial coordinates and their momenta are canonical, while
$$
B_0=\frac{\partial_i\pi_i^*}{M^2},\qquad B_0^*=\frac{\partial_i\pi_i}{M^2}
$$
are dependent operators. Substituting them in the integrated <Hamiltonian> gives the positive density, up to a surface term,
$$
\mathcal H_{\rm red}=\sum_i|\pi_i|^2+\frac{|\partial_i\pi_i|^2}{M^2}+\frac12G_{ik}^*G^{ik}+M^2\sum_i|B^i|^2.
$$
Thus the problem is a constrained choice of canonical variables, not an absence of a consistent free <quantum field theory>.
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