Phyrexian is not a color nor a type of mana, and players cannot add Phyrexian mana. It’s just a symbol that gives you another way to pay for a spell or ability.

A hybrid Phyrexian mana symbol is counted the same way as a normal mana symbol when determining the mana value of a card. Specifically, Tamiyo’s mana value is always 5, even if you pay and 2 life to cast her.

The compleated ability looks only at whether a player chose to pay 2 life for a Phyrexian mana symbol as they were casting the spell. If a player paid life for some other reason while casting the spell, that will not reduce the number of loyalty counters the planeswalker enters the battlefield with.

Compleated is a replacement effect, and it applies only to a permanent that is entering the battlefield with loyalty counters. Any other replacement effect that would apply to the number of loyalty counters it enters the battlefield with will apply as normal.