Per the 出入国在留管理庁 guidelines and document list: resident and national tax certificates, pension and health insurance records, and on-time payment rules to check.
Back within 1 year: use the みなし再入国許可. Longer: get a 再入国許可 in advance (up to 5 years). Conditions, fees, expiry risks abroad, notes for Permanent Residents.
Certificates most used for 在留期間更新 and 永住申請: 住民税 and 国税 certificates, 住民票の写し, pension and health insurance records, 源泉徴収票, 在職証明書: where, how, cost, when.