Final answer:
In a patient with alcoholic hepatitis and portal hypertension demonstrating oliguria and rising BUN, an increased serum creatinine level would be an additional consistent finding, indicating impaired kidney function.
Step-by-step explanation:
A patient with alcoholic hepatitis and portal hypertension showing symptoms of oliguria and increasing blood urea nitrogen (BUN) indicates a decline in kidney function, possibly due to impaired liver function affecting renal circulation. The most consistent additional finding would be an increased serum creatinine, which is another marker for kidney function and would suggest that the kidneys are not filtering waste effectively, a condition often associated with hepatic diseases.