Emergence of String Monodromy in Effective Field Theory

String monodromy is a set of linear relations among open string tree amplitudes with different orderings of the vertex operators. In this Letter, we show how these intrinsically stringy relations emerge in low-energy effective field theory from the assumptions of locality and the field theory Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations. Specifically, we study the bi-adjoint scalar model effective field theory (BAS EFT). We impose the field theory KK and BCJ relations on one of the two color-orderings of the BAS EFT amplitudes and show that the second color-ordering has an emergent set of linear relations that, as checked at 4-point to 36-derivative order, are exactly the monodromy relations. The 4-point results depend on a delicate interplay between consistent factorization of 6-point BAS EFT amplitudes and the 6-point KK BCJ conditions. As a consequence of the analysis, the 4-point Z-theory tree amplitudes are bootstrapped up to a symmetric function in $s,t,u$ whose simple exponentiated form has free parameters that capture all the odd zeta values when matched to Z-theory.