Although some parts of the story are not completely clear to me, the overall situation seems to be a known problem.
Why this can happen
Connecting signals to an unpowered IC (in your case the second FPGA) is usually outside of that IC's specification (with a few exceptions, typically on ICs which are specifically designed for bus isolation - see below). Check that device's datasheet, to see if the input voltage specification mentions Vcc / Vdd, or a specific voltage. If it mentions voltages referenced to Vcc / Vdd then think about what that means, when the device is unpowered.
If you need help interpreting the datasheet of your FPGA, then supply a link to its electrical specifications page, and we can try to find the relevant part for you.
If you do connect signals to such an unpowered IC, then this tends to try to power the unpowered IC through the ESD protection structures on those signal pins of the unpowered IC. However the IC was never intended to be powered through its signal pins, those signals may be unable to supply the necessary current to power the IC (of course, they were never designed to do that) and so the signal voltages may go out-of-specification, the IC may not be powered correctly (since this method of supplying power was never intended) and a variety of incorrect behaviour can be seen. See the questions & answers to these previous topics, for similar problems:
Isolate microcontroller from board and use other one
Unpowered devices on I2C/SPI bus
how to fix it?
Use appropriate bus isolation ICs, which are designed to allow one part of a bus to be unpowered, without affecting other devices on the powered part of the bus. For example, TI and Analog Devices (among many other manufacturers) make such devices, depending on your speed, current, package, cost, availability and other requirements.
In some cases, the 74LCX125 (which has a specification that allows for active input signals, even when it is unpowered) is an example of a type of IC which can be used to buffer signals to an unpowered device i.e. the 74LCX125 and the unpowered device would be connected to the same power rail, and therefore become powered or unpowered together.