I found the following in PRO-SPO-40-50, ABN and EMER Procedures:
Electrical Emergency Configuration
Engine Anti-ice
Engine anti-ice valves are permanently open, although the ECAM memo ENG A.ICE is not displayed on the ECAM (except if the ENG A. ICE pb is at ON).
This confirms what tyropicard said and whilst it's not discussed in the context of the condition you've described, it highlights that there are situations where engine anti-ice can be on but without the relevant status or message to inform the crew. "Increased fuel consumption" is most probably sensed by another means (not the engine anti-ice pb).