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ABSTRACT Jetted astrophysical phenomena with black hole engines, including binary mergers, jetted tidal disruption events, and X-ray binaries, require a large-scale vertical magnetic field for efficient jet formation. However, a dynamo mechanism that could generate these crucial large-scale magnetic fields has not been identified and characterized. We have employed three-dimensional global general relativistic magnetohydrodynamical simulations of accretion discs to quantify, for the first time, a dynamo mechanism that generates large-scale magnetic fields. This dynamo mechanism primarily arises from the non-linear evolution of the magnetorotational instability (MRI). In this mechanism, large non-axisymmetric MRI-amplified shearing wave modes, mediated by the axisymmetric azimuthal magnetic field, generate and sustain the large-scale vertical magnetic field through their non-linear interactions. We identify the advection of magnetic loops as a crucial feature, transporting the large-scale vertical magnetic field from the outer regions to the inner regions of the accretion disc. This leads to a larger characteristic size of the, now advected, magnetic field when compared to the local disc height. We characterize the complete dynamo mechanism with two time-scales: one for the local magnetic field generation, $$t_{\rm gen}$$, and one for the large-scale scale advection, $$t_{\rm adv}$$. Whereas the dynamo we describe is non-linear, we explore the potential of linear mean field models to replicate its core features. Our findings indicate that traditional $$\alpha$$-dynamo models, often computed in stratified shearing box simulations, are inadequate and that the effective large-scale dynamics is better described by the shear current effects or stochastic $$\alpha$$-dynamos.