To sterilize jars, submerge them right-side up in boiling water for 10 minutes.
Pare the ends and peel the bases of the asparagus stalks, then cut into 4-inch lengths to fit jars. If using green garlic, peel the outer layer, then trim and cut into quarters.
Arrange three sterilized wide-mouth pint jars (see note) on a countertop. Into each jar, place a quarter-head of green garlic (or a clove of regular garlic), 1 branch of fresh tarragon, 1/2 a bay leaf, 1/2 a dried red chili, 6 peppercorns, 5 coriander seeds, and 2 whole allespice. Set aside.
Fill a large bowl of water with ice cubes and cold water. Set aside. Blanch the asparagus in a pot of rapidly boiling water until bright green, just about 45 seconds. Transfer to the ice water bath with a slotted spoon. Drain, then pack the asparugus spears snugly into the jars, but don't overcrowd.
In a small saucepan, combine the vinegar, water, sugar and salt. Bring to a boil over high heat, and stir until sugar and salt have dissolved. Fill each jar with the hot vinegar mixture, leaving a 1/2-inch at the top. Wipe the rims with a clean damp cloth, and seal. Finish by processing submerged in boiling water for 10 minutes, starting the 10 minutes when the water returns to a rolling boil after putting the jars in.
Set on a counter to cool, then store in a cool, dark place. Enjoy them immediately, but store in the refrigereator after opening.