Need a better description of what you are trying to do here. While the RX has a SHORT fifo, it is not infinite. At the RX end of a link, there is no relationship between the phase of REFCLK and RXCLK. REFCLK is only used (at the RX end of a link) to ensure that the receive PLL is within the allowed operating range. This is normally around +/- a few hundred PPM. If the received bitstream is outside of the allowed limits, then the receiver will lock to the REFCLK and ignore the serial data stream. If the RX FIFO is enabled (FIFOBYP* is HIGH), this output changes on the rising edge of the RXCLK input.
The description of having data continuous for millions of bytes is more concerned. That is not possible in any FIFO mode to do this, UNLESS the data stream contains JK pairs that can be discarded. If the RX FIFO is bypassed, then you can run continuous, but then the RXCLK is an output.
Also, as far as FIFO flags, there are three flags, and to understand what is going on we need to see the HALF flag too.