Combine 1 cup filtered water, 1/2 tablespoon grated ginger and 1/2 tablespoon sugar in a clean 1-quart jar; stir and loosely cover with a coffee filter or cloth.

For days 2–5 each morning, add 1/2 tablespoon ginger and 1/2 tablespoon sugar to the jar, stir, and cover; by day 5 the bug should be bubbly and yeasty.

Bring 1 liter filtered water to a boil in a large saucepan; remove from heat and add 3 tablespoons chopped ginger, steeping 15–20 minutes off the heat.

While still warm, stir in 1/2 cup sugar until dissolved; add optional spices, steep 5 more minutes, then strain out all solids.

Strain the ginger tea through a fine mesh into a 2-quart jar, pressing solids to extract flavor; allow to cool to room temperature (30–45 minutes).

Stir in 2 tablespoons fresh lemon juice and 1/4 teaspoon cream of tartar into the cooled ginger tea.

Strain 1/2 cup ginger bug liquid through cheesecloth, discard solids, add liquid to the ginger tea and stir for 30 seconds.

Funnel the mixture into bottles, filling each 3/4 full; seal tightly and ferment at room temperature out of sunlight for 48 hours.

After 48 hours, carefully open one bottle to test carbonation; if needed, recork and ferment up to an additional 24 hours, then refrigerate all bottles.