Repeated NONCE in CTR mode
This would be better than simply CTR if your nonces do collide, but it is still not very good.

Suppose you use the same IV for two messages $m_0||m_1||...m_n$ and $m_0'||m_1'||...m_n'$ where all the $m_i$ are 128-bit blocks. Each block is XORed with the same keystream value as the corresponding block in the other message. They then pass through AES, the same as in ECB mode. If any $m_i=m_i'$, they have the same ciphertext, so the cipher leaks information about the message.

SIV mode would be closer to what you are looking for. It should be approximately as fast, though it does require two separate passes over the data. In the case of nonce reuse it leaks the equality of the (complete) messages, but no further information.