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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 followingzero-field cooling and appropriate demagnetization. The path dependency yieldsa variation in the exchange bias depending on the sign of the starting fieldand the path followed in tracing the hysteresis loop. The asymmetry thusobserved - $\Delta$H$_{SEB}$ - is found to be decreasing nonmonotonicallyacross 5-300 K with a peak around 200 K. The $\Delta$H$_{SEB}$ together withlarge H$_{SEB}$ could have significant ramification in tuning the exchange biasdriven effects and consequent applications.

Spontaneous exchange bias in a nanocomposite of BiFeO$_3$-Bi$_2$Fe$_4$O$_9$