Does CPU without register exist? Is existence of registers in CPU an obligation, Or we just do it for performance?

Can we theoretically and practically have computer architectures that don't use registers?
If yes, please name computer architecture and computer models that doesn't have register.

Update: Do stack machines use register? If yes, is it possible to have stack machines without registers?
Every CPU has, at the very least, some internal registers: a program counter, a register storing flags, and in some architectures, an accumulator (though the latter isn't strictly required).

As you mention, registers are important for speed reasons. CPUs are designed with efficiency in mind. There are no drawbacks to having (external) registers, so I conjecture that all CPUs have them.