  We have observed a large as well as path-dependent spontaneous exchange bias
(H$_{SEB}$) ($\sim$30-60 mT) in a nanocomposite of BiFeO$_3$-Bi$_2$Fe$_4$O$_9$
across 5-300 K when it is measured in an unmagnetized state following
zero-field cooling and appropriate demagnetization. The path dependency yields
a variation in the exchange bias depending on the sign of the starting field
and the path followed in tracing the hysteresis loop. The asymmetry thus
observed - $\Delta$H$_{SEB}$ - is found to be decreasing nonmonotonically
across 5-300 K with a peak around 200 K. The $\Delta$H$_{SEB}$ together with
large H$_{SEB}$ could have significant ramification in tuning the exchange bias
driven effects and consequent applications.
