A. Jeff Pitcher, vice president of CP Industries, a manufacturer and distributor of adhesives for the wood-products industry, responds: If not for the stain, any exterior wood glue or exterior-grade adhesive would work fine in this application. But oil-based stains could potentially interfere with the glue line, a problem you could avoid by not using any stain on the gluing surfaces. If that isn't possible, use either a solvent-based construction adhesive, such as PL 400 (Henkel, 800/999-8920, www.stickwithpl.com) or a polyurethane-based construction adhesive, such as Chem-Calk (Bostik, 888/603-8558, www.bostik-us.com).
Avoid the temptation to use a reactive polyurethane glue like Gorilla Glue (800/966-3458, www.gorillaglue.com). While it's true that this type of adhesive can provide a sturdy glue joint in the presence of surface contaminants, its curing process causes it to foam. Because you won't be able to apply adequate clamping pressure until the glue...
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