Ferroelectric switching in BiFeO$_3$ multiferroic thin films with intrinsic ``stripe-like and ``bubble-like polydomain configurations was studied by piezoresponse force microscopy. Using the local electric field applied by a scanning probe microscope tip, we observe reversal of both out-of-plane and in-plane components of the polarization, with the final domain state depending on the tip sweeping direction. In ``bubble-like samples, complete control of the polarization is achieved, with in-plane polarization change mediated and stabilized by out-of-plane polarization reversal. In ``stripe-like samples the intrinsic domain structure influences polarization switching and in-plane reversal may occur without out-of-plane change. The observed switching behaviour can be well correlated with the radial and vertical components of the highly inhomogeneous electric field applied by the tip.
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