  We define an integral Euler cycle for a vector bundle $E$ over an effective
orbifold $X$ for which $(E, X)$ is (stably) normally complex. The
transversality is achieved by using Fukaya-Ono's "normally polynomial
perturbations" and Brett Parker's generalization to "normally complex
perturbations." Two immediate applications in symplectic topology are the
definition of integer-valued genus-zero Gromov--Witten type invariants for
general compact symplectic manifolds using the global Kuranishi chart
constructed by Abouzaid-McLean-Smith, and an alternative proof of the
cohomological splitting theorem of Abouzaid-McLean-Smith for Hamiltonian
fibrations over $S^2$ with integer coefficients.
