Yes there are silent tactile switches. They have dampeners on the stem that cushion the upstroke and the downstroke of the switch. An example of such a switch is the Gazzew Boba U4s.
The dampeners also cushion the bottom out of the keypress so they are a bit easier on the fingers if you are a heavy typist.
In order to deaden the sound further there are various foam products (poron, PE foam, silicon, polyphill) that can be used in the build. There are also mods that can be done like putting painters tape on the back of the PCB or straightening the stabiliser wires and lubing the stabilisers.
It also helps to select a case that doesn't suffer from acoustic resonance.
If you like tactility and silence it might be worth checking out keyboards with silent Topre switches like the HHKB. Although they are not considered a mechanical keyboard some people actually prefer them.
Electrocapacitive switches are what I prefer. They are rubber domes and springs. Even more quiet but somewhat less "tactile" are the boba gum linear switches. They are the quietest mechanical keyswitch I've ever heard and aren't very expensive either.
There are dampener in the shape of O-rings or square pads to make the keyboard more silent. Also, some newer keyboards have a padding in the bottom part to make the sound quieter and more dry.
I love the tactil feedback and key travel but I absolutely hate the sound of mechanical switches (or any loud sound for that matter).
Is there a technical solution out there to have a keyboard with good feedback and travel but near silent?