A dirty chai latte adds espresso to a chai-and-milk drink. Its character depends strongly on the tea preparation: unsweetened spiced tea, a sweetened concentrate and instant chai powder are different starting points. Knowing which one you have helps prevent a drink that is unintentionally too sweet or too diluted.
Prepare the tea according to its instructions and make the espresso separately. Remove tea bags and whole spices before combining the liquids. Add a measured amount of warmed or steamed milk, then taste before sweetening. Cinnamon, cardamom and other spices can be adjusted to preference, but adding more spice does not necessarily make weak coffee taste clearer. Keep the initial recipe simple enough that you can identify what needs changing.
For an iced version, cool the tea and coffee before assembling them with milk and ice in a suitable glass. Account for ice dilution and preserve the same proportions when comparing recipes. If using a pre-sweetened concentrate, reduce or omit extra syrup initially. The goal is a deliberate balance of tea, coffee and milk, without relying on an uncertain anecdote about who first combined them.

The conversation
A good question can make a better brew.