Art De La Guerre

[Black Prince]

If the non phasing player's base is fighting in two directions one enemy base to its front and another on it is flank. The non phasing play destroys the enemy to its front in melee does it automatically conform the enemy base on its flank at the end of the turn?

[Ramses II]

Yes. 
See ‘Conforming after a melee’ p55 and ‘timing of Conformation’ in the FAQ p16. 

[Black Prince]

Thanks it was the statement "this conformation replaces pursuit" had me wondering if only the phasing player could conform.

[Ramses II]

Yup
Units always conform, but can only pursue in their own turn. 

Note also that in this case, as the unit conforms with the enemy unit attacking it’s flank, that enemy may evade (if it has that ability)