The photoionization of hydrogen atoms in a strong magnetic field is formulated as a multichannel problem by representing the asymptotic electron wave function in cylindrical coordinates. Departures from cylindrical symmetry close to the nucleus are incorporated by an $R$-matrix treatment at short range, which then merges with standard quantum-defect procedures. The $R$-matrix calculation utilizes the eigenchannel approach, recast in noniterative form. At the field strength treated here, $B=4.7\ifmmode\times\else\texttimes\fi{}{10}^{9}$ G, the photoionization cross section displays narrow "autoionizing" resonances near the excited Landau thresholds.