If $\tau_1$ and $\tau_2$ meet your requirement, then they are called $T_1$-independent. The following paper may contain useful information: Shakhmatov, D.; Tkachenko, M.; Wilson, R. G. Transversal and $T_1$-independent topologies. Houston J. Math. 30 (2004), no. 2, 421–433. MR2852951.
Lajos Soukup
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