Are there any tasteless edible salts?
Giving a cursory Google e-book search gave me results particularly from old textbooks of 19th century:

Phosphate of soda (Sodium phosphate, $\ce{Na3PO4}$) - was sold under the name of tasteless salt
Sulfate of lime (calcium sulfate, $\ce{CaSO4}$) - it was named agustine because this salt was considered to have no taste. It was considered to mix with acids to form tasteless salts
Phosphates of barytes (barium phosphate, $\ce{Ba3(PO4)2}$) - considered tasteless
Phosphate of strontium (strontium phoshate, $\ce{Sr3(PO4)2}$) - considered tasteless
Superphosphate of lime (calcium superphosphate, $\ce{Ca(H2PO4)2}$) - at high temperature, it melts to a semi-transparent glass which is tasteless.
Sulfate of strontium (strontium sulfate, $\ce{SrSO4}$) - considered tasteless

@Ivan said in a comment(deleted) that calcium sulfate, barium phosphate, strontium phosphate and strontium sulfate are insoluble in water, so they cannot serve the purpose of breaking the emulsion. Finally as OP clarifies, calcium superphosphate is corrosive and hence not safe to eat, it all comes down to sodium phosphate which is both tasteless and edible.
References(in order of mentioned points)

London Medical Gazette: Or, Journal of Practical Medicine, Volume 18, 1836
The Critical Review: Or, Annals of Literature, Volume 33
Tobias Smollett, 1801
Pantologia: A New Cabinet Cyclopaedia, Comprehending a Complete Series of Essays, Treatises, and Systems, Alphabetically Arranged, J. Walker, 1819
The Cyclopædia, Or, Universal Dictionary of Arts, Sciences, and Literature, Volume 39, Longman, Hurst, Rees, Orme &amp; Brown, 1819