Struvite stones (ammonium magnesium phosphate) commonly form staghorn caliculi.
Other clues in the question stem which point to struvite stones:
FA2020 p598 has different types of kidney stones.
Cysteine and uric acid stones are both treated with alkalinization of the urine and more associated with acidic urine. Cysteinuria, though it is associated sometimes with staghorn caliculi, is also hereditary, begins in childhood, and is not associated with bacteriuria. Calcium oxalate and calcium phosphate stones are less often associated with staghorn caliculi.