We present new side-channel attacks on SIKE, the isogeny-based candidate in the NIST PQC competition. Previous works had shown that SIKE is vulnerable to differential power analysis, and pointed to coordinate randomization as an effective countermeasure. We show that coordinate randomization alone is not sufficient, because SIKE is vulnerable to a class of attacks similar to refined power analysis in elliptic curve cryptography, named zero-value attacks. We describe and confirm in the lab two such attacks leading to full key recovery, and analyze their countermeasures.