The colors of mana are white, blue, black, red, and green. Exotic Orchard can't be tapped for colorless mana, even if a land an opponent controls could produce colorless mana.

Exotic Orchard checks the effects of all mana-producing abilities of lands your opponents control, but it doesn't check their costs. For example, Vivid Crag has the ability "{T}, Remove a charge counter from Vivid Crag: Add one mana of any color." If an opponent controls Vivid Crag and you control Exotic Orchard, you can tap Exotic Orchard for any color of mana. It doesn't matter whether Vivid Crag has a charge counter on it, and it doesn't matter whether it's untapped.

When determining what colors of mana your opponents' lands could produce, Exotic Orchard takes into account any applicable replacement effects that would apply to those lands' mana abilities (such as Contamination's effect, for example). If there are more than one, consider them in any possible order.

Exotic Orchard doesn't care about any restrictions or riders your opponents' lands (such as Ancient Ziggurat or Hall of the Bandit Lord) put on the mana they produce. It just cares about colors of mana.

Lands that produce mana based only on what other lands "could produce" won't help each other unless some other land allows one of them to actually produce some type of mana. For example, if you control an Exotic Orchard and your opponent controls an Exotic Orchard and a Reflecting Pool, none of those lands would produce mana if their mana abilities were activated. On the other hand, if you control a Forest and an Exotic Orchard, and your opponent controls an Exotic Orchard and a Reflecting Pool, then each of those lands can be tapped to produce {G}. Your opponent's Exotic Orchard can produce {G} because you control a Forest. Your Exotic Orchard and your opponent's Reflecting Pool can each produce {G} because your opponent's Exotic Orchard can produce {G}.