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](https://doi.org/10.1016/j.topol.2011.07.023)*. Houston J. Math. 30 (2004), no. 2, 421–433. [MR2852951](https://mathscinet.ams.org/mathscinet-getitem?mr=2852951).