  In the one-way model of measurement-based quantum computation (MBQC),
computation proceeds via measurements on some standard resource state.
So-called flow conditions ensure that the overall computation is deterministic
in a suitable sense, with Pauli flow being the most general of these. Existing
work on rewriting MBQC patterns while preserving the existence of flow has
focused on rewrites that reduce the number of qubits.
  In this work, we show that introducing new Z-measured qubits, connected to
any subset of the existing qubits, preserves the existence of Pauli flow.
Furthermore, we give a unique canonical form for stabilizer ZX-diagrams
inspired by recent work of Hu & Khesin. We prove that any MBQC-like stabilizer
ZX-diagram with Pauli flow can be rewritten into this canonical form using only
rules which preserve the existence of Pauli flow, and that each of these rules
can be reversed while also preserving the existence of Pauli flow. Hence we
have complete graphical rewriting for MBQC-like stabilizer ZX-diagrams with
Pauli flow.
