If the tap is large enough you may be successful with a Walton tap extractor

These use 3 blades to engage with the flutes of the broken tap, allowing you to get a grip on it. The blades used to be available as spare parts, as they do tend to get mangled in use. These used to be held in the stores when I worked at Broadbent Machine Tools in the early '80s - handy, as I had something of a tendancy to break taps !

Broken taps can also be drilled out with carbide or Stellite drills, though this can't really be done free hand. Stellite drills especially rely upon heavy axial pressure to get them to "cut" - which isn't really cutting at all, more heating the material to be drilled by friction to soften it & allow it to be scraped off. I have had some success regrinding carbide masonary bits to metal cutting geometry of one-off jobs drilling hard bits - can't recall the specific jobs, but used slowly with a constant pressure and something like Rocol RTD cutting fluid, so probabaly a work-hardening material being drilled.

It is also possible to dissovle broken taps using a super saturated solution of alum (aluminium potassium sulphate, I believe). This method can be used if the tap is broken off in a non-ferrous material as it causes rapid oxidation of the steel tap - obviously it would also cause rapid oxidation of any ferrous metal the tap was broken off in as well. Read of this quite a bit, but never tried it.

I have occasionally had success in shattering a broken tap using a punch & then extracting the bits, though this can cause some damage to the hole.

Are the tank badge threads in chromed tanks ? Chrome plating in the threads can be hard on taps. Trying a new HSS tap from a reputable manufacturer would be something to try first ?

Nigel B.