A wiring diagram is an arrangement of $m$ pseudolines, every two of which cross exactly once. Reading its crossings from left to right encodes it as a reduced word of the permutation $w_0\in S_m$ that reverses $1,\dots,m$, unique up to commutations, and a braid move $σ_iσ_{i+1}σ_i\leftrightarrowσ_{i+1}σ_iσ_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\operatorname{VC}(G)$, where $\operatorname{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding distance in the mutation graph is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge4$, and by duality in every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge2$.

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