Multiply using ***Real*** numbers

Python,  54  52 bytes

lambda a,b:[v for v in(a,b,a*b%-~99)if'd'>chr(v)][2]

An unnamed function raising an error if either of the inputs are fake otherwise returning the product of the inputs modulo 100 (always real and correct if the product itself is real).

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How?

First creates a tuple of the inputs and their product modulo 100 using (a,b,a*b%-~99) where ~x computes -1-x and % is the modulo operator.

Then traverses that tuple creating a list of those values which are real numbers.

The test for real-ness is performed by first attempting to cast each value, v, to characters with chr(v) - which will raise a ValueError if v is not a non-negative integer less than 256, then checking the value is less than 100 by a less-than-comparison, <, to the 100^th character, 'd'.

If both a and b are real the list will have three entries (the product modulo 100 is alway real), otherwise it will have less. The [...][2] attempts to get the item at the second index (third item), raising an IndexError if it does not exist.

Thus either input being fake results in either a ValueError or IndexError, while both inputs being real always succeeds and returns a real number, and if that product is itself real it is the returned value.

Python 2, 41 38 41 39 bytes

lambda a,b:a*b%-~99/(chr(a)<'d'>chr(b))

-2 bytes from xnor.

Uses -~99 to get 100 legally.

Relies on u<v>w is equivalent to u<v and v>w.

If a or b are not integers chr throws a TypeError. If either are less than 0, chr throws a ValueError.

If a or b are greater than or equal to 100 (ord('d')), then the denominator will evaluate to 0, and throws a ZeroDivisionError.

Otherwise, the denominator will evaluate to 1; and returns the modded product.